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Theorem ovmpodx 7539
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpodx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpodx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpodx.3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpodx.4 (𝜑𝐴𝐶)
ovmpodx.5 (𝜑𝐵𝐿)
ovmpodx.6 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpodx (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐴   𝑥,𝐵   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpodx
StepHypRef Expression
1 ovmpodx.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpodx.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 ovmpodx.3 . 2 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
4 ovmpodx.4 . 2 (𝜑𝐴𝐶)
5 ovmpodx.5 . 2 (𝜑𝐵𝐿)
6 ovmpodx.6 . 2 (𝜑𝑆𝑋)
7 nfv 1917 . 2 𝑥𝜑
8 nfv 1917 . 2 𝑦𝜑
9 nfcv 2902 . 2 𝑦𝐴
10 nfcv 2902 . 2 𝑥𝐵
11 nfcv 2902 . 2 𝑥𝑆
12 nfcv 2902 . 2 𝑦𝑆
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpodxf 7538 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  (class class class)co 7390  cmpo 7392
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 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  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 6481  df-fun 6531  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395
This theorem is referenced by:  ovmpod  7540  ovmpox  7541  dpjfval  19881  fgval  23298  om1val  24470  pi1val  24477  dvfval  25338  dvnfval  25363  taylfval  25795  line  47052  rrxline  47054
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