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Theorem ovmpt2a 5683
Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2ga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2ga.2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpt2a.4 𝑆 ∈ V
Assertion
Ref Expression
ovmpt2a ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem ovmpt2a
StepHypRef Expression
1 ovmpt2a.4 . 2 𝑆 ∈ V
2 ovmpt2ga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
3 ovmpt2ga.2 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
42, 3ovmpt2ga 5682 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
51, 4mp3an3 1258 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2610  (class class class)co 5564  cmpt2 5566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569
This theorem is referenced by: (None)
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