Theorem mpteq2ia

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013)

	Ref	Expression
Hypothesis	mpteq2ia.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion	mpteq2ia	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		mpteq2ia.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
2		eqid	⊢ 𝐴 = 𝐴
3	2	ax-gen	⊢ ∀ 𝑥 𝐴 = 𝐴
4	1	rgen	⊢ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶
5		mpteq12f	⊢ ( ( ∀ 𝑥 𝐴 = 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
6	3 4 5	mp2an	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )