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Theorem ovmpt2x2 42644
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6935 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
ovmpt2x2.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2x2.2 (𝑦 = 𝐵𝐶 = 𝐿)
ovmpt2x2.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2x2 ((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 ovmpt2x2
StepHypRef Expression
1 ovmpt2x2.3 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21a1i 11 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
3 ovmpt2x2.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
43adantl 474 . 2 (((𝐴𝐿𝐵𝐷𝑆𝐻) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
5 ovmpt2x2.2 . . 3 (𝑦 = 𝐵𝐶 = 𝐿)
65adantl 474 . 2 (((𝐴𝐿𝐵𝐷𝑆𝐻) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
7 simp1 1128 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐴𝐿)
8 simp2 1129 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝐵𝐷)
9 simp3 1130 . 2 ((𝐴𝐿𝐵𝐷𝑆𝐻) → 𝑆𝐻)
102, 4, 6, 7, 8, 9ovmpt2rdx 42643 1 ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1069   = wceq 1629  wcel 2143  (class class class)co 6791  cmpt2 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796
This theorem is referenced by:  lincval  42723
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