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Theorem ovmpod 5995
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
ovmpod.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpod.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpod.3 (𝜑𝐴𝐶)
ovmpod.4 (𝜑𝐵𝐷)
ovmpod.5 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpod (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpod
StepHypRef Expression
1 ovmpod.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpod.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 eqidd 2178 . 2 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐷)
4 ovmpod.3 . 2 (𝜑𝐴𝐶)
5 ovmpod.4 . 2 (𝜑𝐵𝐷)
6 ovmpod.5 . 2 (𝜑𝑆𝑋)
71, 2, 3, 4, 5, 6ovmpodx 5994 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  (class class class)co 5868  cmpo 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873
This theorem is referenced by:  ovmpoga  5997  iseqovex  10429  seqvalcd  10432  resqrexlemp1rp  10986  resqrexlemfp1  10989  lcmval  12033  ennnfonelemg  12374  plusfvalg  12661  grpsubval  12796  mulgval  12862  cnfval  13327  cnpfval  13328  blvalps  13521  blval  13522
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