LemmaStatement: Literature-sourcedProof: 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.

Functions $f$ and $g$ are equal if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that common domain

Statement

Let $f$ and $g$ be functions. Then $f = g$ if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x \in dom f$ .

Facts & Assumptions

Given: functions $f$ and $g$ .

[L1]

A function is a relation $f$ such that $(a, b) \in f$ and $(a, c) \in f$ imply $b = c$ (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain).

[L2]

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

[L3]

$(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ( $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ).

[L4]

If every $z$ satisfies $z \in x$ if and only if $z \in y$ , then $x = y$ (The Axiom of Extensionality: $\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y)$ ).

[L5]

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

Proof

technique · direct

1.1L1L2L4

From left to right: if $f = g$ then the two sets have the same elements, so the defining conditions for the domain and for the value at a point are the same for both.

1.2L1L2L3L4L5

From right to left: assume $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that set. Let $z \in f$ . Then $z$ is an ordered pair $(x, y)$ with $x \in dom f$ and $y = f (x)$ ; hence $x \in dom g$ and $y = g (x)$ , so $z \in g$ . The same argument with $f$ and $g$ exchanged gives the reverse inclusion, so the two sets have the same elements.

Functions $f$ and $g$ are equal if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that common domain

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Functions ff and gg are equal if and only if dom⁡f=dom⁡gdomf=domg and f(x)=g(x)f(x)=g(x) for every xx in that common domain

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Functions $f$ and $g$ are equal if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that common domain