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Theorem ovmpox 7282
 Description: The value of an operation class abstraction. Variant of ovmpoga 7283 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
ovmpox.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpox.2 (𝑥 = 𝐴𝐷 = 𝐿)
ovmpox.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpox ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐿,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpox
StepHypRef Expression
1 elex 3459 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpox.3 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 11 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpox.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 485 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 ovmpox.2 . . . 4 (𝑥 = 𝐴𝐷 = 𝐿)
76adantl 485 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8 simp1 1133 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐴𝐶)
9 simp2 1134 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐵𝐿)
10 simp3 1135 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝑆 ∈ V)
113, 5, 7, 8, 9, 10ovmpodx 7280 . 2 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
121, 11syl3an3 1162 1 ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  (class class class)co 7135   ∈ cmpo 7137 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140 This theorem is referenced by:  evls1fval  20943  ptbasfi  22186  tglngval  26345  extdgval  31132  scutval  33378  igenval  35499  lcoop  44818
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