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Theorem ovmpt2x 6687
 Description: The value of an operation class abstraction. Variant of ovmpt2ga 6688 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 3185 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpt2x.3 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 11 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpt2x.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 481 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 ovmpt2x.2 . . . 4 (𝑥 = 𝐴𝐷 = 𝐿)
76adantl 481 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8 simp1 1054 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐴𝐶)
9 simp2 1055 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐵𝐿)
10 simp3 1056 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝑆 ∈ V)
113, 5, 7, 8, 9, 10ovmpt2dx 6685 . 2 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
121, 11syl3an3 1353 1 ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  evls1fval  19505  ptbasfi  21194  tglngval  25246  igenval  33030  lcoop  41994
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