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Theorem mpoeq123i 7485
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 7484 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87mptru 1549 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wtru 1543  cmpo 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  ofmres  7971  seqval  13977  oppgtmd  23601  wlkson  28913  mdetlap1  32806  sdc  36612  tgrpset  39616  mendvscafval  41932  fsovcnvlem  42764  hspmbl  45345
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