Theorem mpoeq123i 7425

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 7424	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
8	7	mptru 1547	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ cmpo 7351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-oprab 7353  df-mpo 7354

This theorem is referenced by:  ofmres  7919  seqval  13919  oppgtmd  23982  seqsval  28187  wlkson  29600  mdetlap1  33799  sdc  37734  tgrpset  40734  mendvscafval  43169  fsovcnvlem  43996  hspmbl  46620  setc1ocofval  49489