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Theorem ovmpog 7518
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 2793 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
4 ovmpog.3 . 2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
53, 4ovmpoga 7513 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  (class class class)co 7361  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366
This theorem is referenced by:  ovmpo  7519  naddcllem  8626  mapvalg  8781  pmvalg  8782  genpv  10943  shftfval  14964  efmndov  18699  frlmipval  21208  bcthlem1  24711  negsval  27353  motplusg  27533  signspval  33228  elghomlem1OLD  36394  paddval  38311  tgrpov  39261  erngmul  39319  erngmul-rN  39327  dvamulr  39525  dvavadd  39528  dvhmulr  39599  djavalN  39648  djhval  39911  mendmulr  41562
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