Theorem mpoeq12

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

	Ref	Expression
Assertion	mpoeq12	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐸 = 𝐸
2	1	rgenw	⊢ ∀ 𝑦 ∈ 𝐵 𝐸 = 𝐸
3	2	jctr	⊢ ( 𝐵 = 𝐷 → ( 𝐵 = 𝐷 ∧ ∀ 𝑦 ∈ 𝐵 𝐸 = 𝐸 ) )
4	3	ralrimivw	⊢ ( 𝐵 = 𝐷 → ∀ 𝑥 ∈ 𝐴 ( 𝐵 = 𝐷 ∧ ∀ 𝑦 ∈ 𝐵 𝐸 = 𝐸 ) )
5		mpoeq123	⊢ ( ( 𝐴 = 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐵 = 𝐷 ∧ ∀ 𝑦 ∈ 𝐵 𝐸 = 𝐸 ) ) → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) )
6	4 5	sylan2	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝐸 ) )