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Theorem mpoeq3dv 5935
Description: An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)
Hypothesis
Ref Expression
mpoeq3dv.1 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
mpoeq3dv (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem mpoeq3dv
StepHypRef Expression
1 mpoeq3dv.1 . . 3 (𝜑𝐶 = 𝐷)
213ad2ant1 1018 . 2 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
32mpoeq3dva 5933 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cmpo 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-oprab 5873  df-mpo 5874
This theorem is referenced by: (None)
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