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Theorem ovmpoga 7554
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 3485 . 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 481 . . 3 (((𝐴𝐶𝐵𝐷𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 simp1 1133 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐴𝐶)
7 simp2 1134 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐵𝐷)
8 simp3 1135 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝑆 ∈ V)
93, 5, 6, 7, 8ovmpod 7552 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
101, 9syl3an3 1162 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  (class class class)co 7401  cmpo 7403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406
This theorem is referenced by:  ovmpoa  7555  ovmpog  7559  elovmpo  7644  offval  7672  offval3  7962  mptmpoopabbrdOLDOLD  8063  bropopvvv  8070  reps  14717  hashbcval  16934  setsvalg  17098  ressval  17175  restval  17371  sylow1lem4  19511  sylow3lem2  19538  sylow3lem3  19539  lsmvalx  19549  mvrfval  21850  opsrval  21911  marrepfval  22384  marrepval0  22385  marepvfval  22389  marepvval0  22390  cnmpt12  23493  cnmpt22  23500  qtopval  23521  flimval  23789  fclsval  23834  ucnval  24104  stdbdmetval  24345  fldgenval  32868  resvval  32907  irngval  33229  minplyval  33246  ofcfval3  33589  fmulcl  44782
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