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Theorem ovmpog 7566
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
ovmpog.1 (𝑥 = 𝐴𝑅 = 𝐺)
ovmpog.2 (𝑦 = 𝐵𝐺 = 𝑆)
ovmpog.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpog ((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 ovmpog
StepHypRef Expression
1 ovmpog.1 . . 3 (𝑥 = 𝐴𝑅 = 𝐺)
2 ovmpog.2 . . 3 (𝑦 = 𝐵𝐺 = 𝑆)
31, 2sylan9eq 2792 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
4 ovmpog.3 . 2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
53, 4ovmpoga 7561 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  (class class class)co 7408  cmpo 7410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413
This theorem is referenced by:  ovmpo  7567  naddcllem  8674  mapvalg  8829  pmvalg  8830  genpv  10993  shftfval  15016  efmndov  18761  frlmipval  21333  bcthlem1  24840  negsval  27497  motplusg  27790  signspval  33558  elghomlem1OLD  36748  paddval  38664  tgrpov  39614  erngmul  39672  erngmul-rN  39680  dvamulr  39878  dvavadd  39881  dvhmulr  39952  djavalN  40001  djhval  40264  mendmulr  41920
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