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Theorem mpoeq123i 7487
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 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)

Proof of Theorem mpoeq123i
StepHypRef Expression
1 mpoeq123i.1 . . . 4 𝐴 = 𝐷
21a1i 11 . . 3 (⊤ → 𝐴 = 𝐷)
3 mpoeq123i.2 . . . 4 𝐵 = 𝐸
43a1i 11 . . 3 (⊤ → 𝐵 = 𝐸)
5 mpoeq123i.3 . . . 4 𝐶 = 𝐹
65a1i 11 . . 3 (⊤ → 𝐶 = 𝐹)
72, 4, 6mpoeq123dv 7486 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87mptru 1548 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wtru 1542  cmpo 7413
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  ofmres  7973  seqval  13979  oppgtmd  23608  wlkson  28951  mdetlap1  32875  sdc  36698  tgrpset  39702  mendvscafval  42014  fsovcnvlem  42846  hspmbl  45424
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