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Theorem ovmpt2x 5656
 Description: The value of an operation class abstraction. Variant of ovmpt2ga 5657 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
ovmpt2x.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2x.2 (𝑥 = 𝐴𝐷 = 𝐿)
ovmpt2x.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2x ((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 ovmpt2x
StepHypRef Expression
1 elex 2583 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpt2x.3 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 9 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpt2x.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 266 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 ovmpt2x.2 . . . 4 (𝑥 = 𝐴𝐷 = 𝐿)
76adantl 266 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8 simp1 915 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐴𝐶)
9 simp2 916 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐵𝐿)
10 simp3 917 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝑆 ∈ V)
113, 5, 7, 8, 9, 10ovmpt2dx 5654 . 2 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
121, 11syl3an3 1181 1 ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  Vcvv 2574  (class class class)co 5539   ↦ cmpt2 5541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544 This theorem is referenced by: (None)
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