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Theorem ovmpoga 7561
Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpoga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpoga.2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpoga ((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)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpoga
StepHypRef Expression
1 elex 3492 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpoga.2 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 11 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpoga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 482 . . 3 (((𝐴𝐶𝐵𝐷𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 simp1 1136 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐴𝐶)
7 simp2 1137 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐵𝐷)
8 simp3 1138 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝑆 ∈ V)
93, 5, 6, 7, 8ovmpod 7559 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
101, 9syl3an3 1165 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  (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:  ovmpoa  7562  ovmpog  7566  elovmpo  7650  offval  7678  offval3  7968  mptmpoopabbrdOLD  8068  bropopvvv  8075  reps  14719  hashbcval  16934  setsvalg  17098  ressval  17175  restval  17371  sylow1lem4  19468  sylow3lem2  19495  sylow3lem3  19496  lsmvalx  19506  mvrfval  21539  opsrval  21600  marrepfval  22061  marrepval0  22062  marepvfval  22066  marepvval0  22067  cnmpt12  23170  cnmpt22  23177  qtopval  23198  flimval  23466  fclsval  23511  ucnval  23781  stdbdmetval  24022  fldgenval  32397  resvval  32436  irngval  32744  minplyval  32761  ofcfval3  33095  fmulcl  44287
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