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Theorem mpt2eq3ia 5597
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpt2eq3ia.1 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2eq3ia (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Proof of Theorem mpt2eq3ia
StepHypRef Expression
1 mpt2eq3ia.1 . . . 4 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
213adant1 933 . . 3 ((⊤ ∧ 𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
32mpt2eq3dva 5596 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
43trud 1268 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wtru 1260  wcel 1409  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-oprab 5543  df-mpt2 5544
This theorem is referenced by:  oprab2co  5866  genpdf  6663  dfioo2  8943
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