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	⊢ ( 𝜑 → Fun 𝐹 )
	bnj1502.2	⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 )
	bnj1502.3	⊢ ( 𝜑 → 𝐴 ∈ dom 𝐺 )
Assertion	bnj1502	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1502.1	⊢ ( 𝜑 → Fun 𝐹 )
2		bnj1502.2	⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 )
3		bnj1502.3	⊢ ( 𝜑 → 𝐴 ∈ dom 𝐺 )
4		funssfv	⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )