Theorem bnj1502

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1502.1	$$⊢\phi \to \mathrm{Fun}F$$
	bnj1502.2	$$⊢\phi \to G\subseteq F$$
	bnj1502.3	$$⊢\phi \to A\in \mathrm{dom}G$$
Assertion	bnj1502	$$⊢\phi \to F\left(A\right)=G\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1502.1	$$⊢\phi \to \mathrm{Fun}F$$
2		bnj1502.2	$$⊢\phi \to G\subseteq F$$
3		bnj1502.3	$$⊢\phi \to A\in \mathrm{dom}G$$
4		funssfv	$$⊢\left(\mathrm{Fun}F\wedge G\subseteq F\wedge A\in \mathrm{dom}G\right)\to F\left(A\right)=G\left(A\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to F\left(A\right)=G\left(A\right)$$