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Theorem ovmpodx 7285
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 1915 . 2 𝑥𝜑
8 nfv 1915 . 2 𝑦𝜑
9 nfcv 2979 . 2 𝑦𝐴
10 nfcv 2979 . 2 𝑥𝐵
11 nfcv 2979 . 2 𝑥𝑆
12 nfcv 2979 . 2 𝑦𝑆
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpodxf 7284 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  (class class class)co 7140  cmpo 7142
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145
This theorem is referenced by:  ovmpod  7286  ovmpox  7287  dpjfval  19168  fgval  22473  om1val  23633  pi1val  23640  dvfval  24498  dvnfval  24523  taylfval  24952  line  45085  rrxline  45087
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