LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passverified 2026-08-06 (claude-opus-5)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For relations $R$ and $S$ and a set $A$ , the collections ${(b, a) : (a, b) \in R}$ , ${(a, c) : \exists b ((a, b) \in R \land (b, c) \in S)}$ and ${(a, b) \in R : a \in A}$ are sets

Statement

Let $R$ and $S$ be relations and $A$ a set. Then each of

${(b, a) : (a, b) \in R}, {(a, c) : \exists b ((a, b) \in R \land (b, c) \in S)}, {(a, b) \in R : a \in A}$

is a set; the first is included in $ran R \times dom R$ , the second in $dom R \times ran S$ , and the third in $R$ .

Facts & Assumptions

Given: relations $R$ and $S$ and a set $A$ .

[L1]

A relation is a set $R$ every element of which is an ordered pair (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L2]

$dom R : = {a : \exists b (a, b) \in R}, ran R : = {b : \exists a (a, b) \in R}$ (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L3]

$z \in A \times B$ holds if and only if $z = (a, b)$ for some $a \in A$ and some $b \in B$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ).

[L4]

For any parameters $\overset{ˉ}{p}$ and any set $x$ , there is a set $y$ whose elements are exactly the elements $z$ of $x$ for which $φ (z, \overset{ˉ}{p})$ holds (The Axiom Schema of Separation: for each formula $φ$ , $\forall \overset{ˉ}{p} \forall x \exists y \forall z (z \in y \leftrightarrow (z \in x \land φ (z, \overset{ˉ}{p})))$ ).

[L5]

$R \subseteq dom R \times ran R$ (Every relation $R$ satisfies $R \subseteq dom R \times ran R$ , and $R$ is a relation from $A$ to $B$ if and only if $dom R \subseteq A$ and $ran R \subseteq B$ ).

[L6]

$x \subseteq y$ means that every element of $x$ is an element of $y$ (Subset $x \subseteq y$ , proper subset $x ⊊ y$ , and the separation notation ${z \in x : φ (z)}$ ).

[L7]

$(a, b) : = {{a}, {a, b}}$ (The Kuratowski ordered pair $(a, b) : = {{a}, {a, b}}$ ).

Proof

technique · direct

1.1

If $(a, b) \in R$ then $a \in dom R$ and $b \in ran R$ , so $(b, a) \in ran R \times dom R$ ; separating inside that product with the formula $\exists a \exists b (z = (b, a) \land (a, b) \in R)$ and the parameter $R$ therefore yields exactly the first collection.

L2L3L4L6L7

1.2

If $(a, b) \in R$ and $(b, c) \in S$ then $a \in dom R$ and $c \in ran S$ , so $(a, c) \in dom R \times ran S$ ; separating inside that product with the formula $\exists a \exists c (z = (a, c) \land \exists b ((a, b) \in R \land (b, c) \in S))$ and the parameters $R$ and $S$ yields exactly the second collection.

L2L3L4L6L7

1.3

The third collection consists of elements of $R$ , and separating inside $R$ with the formula $\exists a \exists b (z = (a, b) \land a \in A)$ and the parameters $A$ and $R$ yields exactly it; every element of $R$ is an ordered pair, so the formula selects the intended pairs.

L1L4L5L6L7

2.1

All three collections are therefore sets, with the stated inclusions.

step 1.1step 1.2step 1.3∎

Depends on

Used by

The inverse relation R⁻¹, the composite S ∘ R, and the restriction R ↾ A Definition

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

B. Kaya, MATH 320 Set Theory (METU), Def. 9, Def. 12 and Def. 19 (standard reference, not scraped)
Composition of relations (Wikipedia) (standard reference, not scraped)
Binary relation (Wikipedia) (standard reference, not scraped)

For relations RR and SS and a set AA, the collections { (b,a):(a,b)∈R }{(b,a):(a,b)∈R}, { (a,c):∃b ((a,b)∈R∧(b,c)∈S) }{(a,c):∃b ((a,b)∈R∧(b,c)∈S)} and { (a,b)∈R:a∈A }{(a,b)∈R:a∈A} are sets