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Theorem ovmpt2g 5779
 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 2140 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
4 ovmpt2g.3 . 2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
53, 4ovmpt2ga 5774 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 924   = wceq 1289   ∈ wcel 1438  (class class class)co 5652   ↦ cmpt2 5654 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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353 This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657 This theorem is referenced by:  ovmpt2  5780  oav  6215  omv  6216  oeiv  6217  mapvalg  6415  pmvalg  6416  mulpipq2  6930  genipv  7068  genpelxp  7070  subval  7674  divvalap  8141  cnref1o  9133  modqval  9731  frecuzrdgrrn  9815  frec2uzrdg  9816  frecuzrdgrcl  9817  frecuzrdgsuc  9821  frecuzrdgrclt  9822  frecuzrdgg  9823  frecuzrdgsuctlem  9830  iseqval  9871  iseqvalt  9873  seq3val  9874  iseqfclt  9879  seqf  9880  iseqp1  9882  iseqp1t  9883  seq3p1  9884  exp3val  9957  bcval  10157  shftfvalg  10252  shftfval  10255  cnrecnv  10344  gcdval  11229  sqpweven  11431  2sqpwodd  11432  cncfval  11628
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