Theorem mpoeq123i 7436

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 11	. . 3 ⊢ (⊤ → 𝐴 = 𝐷)
3		mpoeq123i.2	. . . 4 ⊢ 𝐵 = 𝐸
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝐵 = 𝐸)
5		mpoeq123i.3	. . . 4 ⊢ 𝐶 = 𝐹
6	5	a1i 11	. . 3 ⊢ (⊤ → 𝐶 = 𝐹)
7	2, 4, 6	mpoeq123dv 7435	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
8	7	mptru 1549	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)

Syntax hints:   = wceq 1542  ⊤wtru 1543   ∈ cmpo 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  ofmres  7930  seqval  13939  oppgtmd  24045  seqsval  28269  wlkson  29711  mdetlap1  33964  sdc  37916  tgrpset  41042  mendvscafval  43464  fsovcnvlem  44290  hspmbl  46909  setc1ocofval  49775