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Theorem ovmpox 5910
 Description: The value of an operation class abstraction. Variant of ovmpoga 5911 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 2701 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpox.3 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 9 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpox.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 275 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 ovmpox.2 . . . 4 (𝑥 = 𝐴𝐷 = 𝐿)
76adantl 275 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8 simp1 982 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐴𝐶)
9 simp2 983 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐵𝐿)
10 simp3 984 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝑆 ∈ V)
113, 5, 7, 8, 9, 10ovmpodx 5908 . 2 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
121, 11syl3an3 1252 1 ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  Vcvv 2690  (class class class)co 5785   ∈ cmpo 5787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-iota 5099  df-fun 5136  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790 This theorem is referenced by:  reldvg  12890
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