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Theorem mpoeq123i 7204
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 7203 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
87mptru 1545 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wtru 1539  cmpo 7132
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-oprab 7134  df-mpo 7135
This theorem is referenced by:  ofmres  7660  seqval  13363  oppgtmd  22681  wlkson  27425  mdetlap1  31102  sdc  35068  tgrpset  37927  mendvscafval  39945  fsovcnvlem  40526  hspmbl  43091
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