Theorem mpoeq123i 5905

Description: An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)

Hypotheses

Ref	Expression
mpoeq123i.1	⊢ 𝐴 = 𝐷
mpoeq123i.2	⊢ 𝐵 = 𝐸
mpoeq123i.3	⊢ 𝐶 = 𝐹

Assertion

Ref	Expression
mpoeq123i	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)

Proof of Theorem mpoeq123i

Step	Hyp	Ref	Expression
1		mpoeq123i.1	. . . 4 ⊢ 𝐴 = 𝐷
2	1	a1i 9	. . 3 ⊢ (⊤ → 𝐴 = 𝐷)
3		mpoeq123i.2	. . . 4 ⊢ 𝐵 = 𝐸
4	3	a1i 9	. . 3 ⊢ (⊤ → 𝐵 = 𝐸)
5		mpoeq123i.3	. . . 4 ⊢ 𝐶 = 𝐹
6	5	a1i 9	. . 3 ⊢ (⊤ → 𝐶 = 𝐹)
7	2, 4, 6	mpoeq123dv 5904	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
8	7	mptru 1352	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)

Syntax hints:   = wceq 1343  ⊤wtru 1344   ∈ cmpo 5844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ofmres  6104