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Theorem ovmpox2 47984
Description: The value of an operation class abstraction. Variant of ovmpoga 7600 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
ovmpox2.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpox2.2 (𝑦 = 𝐵𝐶 = 𝐿)
ovmpox2.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpox2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝑥,𝐿,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem ovmpox2
StepHypRef Expression
1 ovmpox2.3 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21a1i 11 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
3 ovmpox2.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
43adantl 481 . 2 (((𝐴𝐿𝐵𝐷𝑆𝐻) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
5 ovmpox2.2 . . 3 (𝑦 = 𝐵𝐶 = 𝐿)
65adantl 481 . 2 (((𝐴𝐿𝐵𝐷𝑆𝐻) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
7 simp1 1136 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐴𝐿)
8 simp2 1137 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐵𝐷)
9 simp3 1138 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝑆𝐻)
102, 4, 6, 7, 8, 9ovmpordx 47983 1 ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2103  (class class class)co 7445  cmpo 7447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450
This theorem is referenced by:  lincval  48057
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