Theorem funALTVeq

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

	Ref	Expression
Assertion	funALTVeq	⊢ ( 𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵 ) )

Proof

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