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Theorem ovmpoga 7513
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 3465 . 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 483 . . 3 (((𝐴𝐶𝐵𝐷𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 simp1 1137 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐴𝐶)
7 simp2 1138 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐵𝐷)
8 simp3 1139 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝑆 ∈ V)
93, 5, 6, 7, 8ovmpod 7511 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
101, 9syl3an3 1166 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447  (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:  ovmpoa  7514  ovmpog  7518  elovmpo  7602  offval  7630  offval3  7919  mptmpoopabbrdOLD  8019  bropopvvv  8026  reps  14667  hashbcval  16882  setsvalg  17046  ressval  17123  restval  17316  sylow1lem4  19391  sylow3lem2  19418  sylow3lem3  19419  lsmvalx  19429  mvrfval  21412  opsrval  21470  marrepfval  21932  marrepval0  21933  marepvfval  21937  marepvval0  21938  cnmpt12  23041  cnmpt22  23048  qtopval  23069  flimval  23337  fclsval  23382  ucnval  23652  stdbdmetval  23893  fldgenval  32135  resvval  32172  irngval  32423  minplyval  32436  ofcfval3  32765  fmulcl  43912
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