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Theorem ovmpt2g 5714
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovmpt2g.1 (𝑥 = 𝐴𝑅 = 𝐺)
ovmpt2g.2 (𝑦 = 𝐵𝐺 = 𝑆)
ovmpt2g.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2g ((𝐴𝐶𝐵𝐷𝑆𝐻) → (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)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpt2g
StepHypRef Expression
1 ovmpt2g.1 . . 3 (𝑥 = 𝐴𝑅 = 𝐺)
2 ovmpt2g.2 . . 3 (𝑦 = 𝐵𝐺 = 𝑆)
31, 2sylan9eq 2135 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
4 ovmpt2g.3 . 2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
53, 4ovmpt2ga 5709 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5591  cmpt2 5593
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 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316
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 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596
This theorem is referenced by:  ovmpt2  5715  oav  6147  omv  6148  oeiv  6149  mapvalg  6345  pmvalg  6346  mulpipq2  6833  genipv  6971  genpelxp  6973  subval  7577  divvalap  8039  cnref1o  9028  modqval  9620  frecuzrdgrrn  9704  frec2uzrdg  9705  frecuzrdgrcl  9706  frecuzrdgsuc  9710  frecuzrdgrclt  9711  frecuzrdgg  9712  frecuzrdgsuctlem  9719  iseqval  9749  iseqvalt  9751  iseqfclt  9755  iseqp1  9757  iseqp1t  9758  expival  9794  bcval  9992  shftfvalg  10080  shftfval  10083  cnrecnv  10171  gcdval  10731  sqpweven  10933  2sqpwodd  10934
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