Theorem bnj1503

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

	Ref	Expression
Hypotheses	bnj1503.1	$$⊢\phi \to \mathrm{Fun}F$$
	bnj1503.2	$$⊢\phi \to G\subseteq F$$
	bnj1503.3	$$⊢\phi \to A\subseteq \mathrm{dom}G$$
Assertion	bnj1503	$$⊢\phi \to {F\upharpoonright }_A={G\upharpoonright }_A$$

Proof

Step	Hyp	Ref	Expression
1		bnj1503.1	$$⊢\phi \to \mathrm{Fun}F$$
2		bnj1503.2	$$⊢\phi \to G\subseteq F$$
3		bnj1503.3	$$⊢\phi \to A\subseteq \mathrm{dom}G$$
4		fun2ssres	$$⊢\left(\mathrm{Fun}F\wedge G\subseteq F\wedge A\subseteq \mathrm{dom}G\right)\to {F\upharpoonright }_A={G\upharpoonright }_A$$
5	1 2 3 4	syl3anc	$$⊢\phi \to {F\upharpoonright }_A={G\upharpoonright }_A$$