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Theorem mpoeq3ia 5883
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpoeq3ia.1 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpoeq3ia (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Proof of Theorem mpoeq3ia
StepHypRef Expression
1 mpoeq3ia.1 . . . 4 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
213adant1 1000 . . 3 ((⊤ ∧ 𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
32mpoeq3dva 5882 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
43mptru 1344 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wtru 1336  wcel 2128  cmpo 5823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-oprab 5825  df-mpo 5826
This theorem is referenced by:  mpodifsnif  5911  mposnif  5912  oprab2co  6162  genpdf  7423  dfioo2  9873  iseqvalcbv  10351  divcnap  12942
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