Theorem funeq

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	funeq	⊢ ( 𝐴 = 𝐵 → ( Fun 𝐴 ↔ Fun 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐴 )
2		funss	⊢ ( 𝐵 ⊆ 𝐴 → ( Fun 𝐴 → Fun 𝐵 ) )
3	1 2	syl	⊢ ( 𝐴 = 𝐵 → ( Fun 𝐴 → Fun 𝐵 ) )
4		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
5		funss	⊢ ( 𝐴 ⊆ 𝐵 → ( Fun 𝐵 → Fun 𝐴 ) )
6	4 5	syl	⊢ ( 𝐴 = 𝐵 → ( Fun 𝐵 → Fun 𝐴 ) )
7	3 6	impbid	⊢ ( 𝐴 = 𝐵 → ( Fun 𝐴 ↔ Fun 𝐵 ) )