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Theorem ovig 7355
Description: The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovig.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
ovig.2 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
ovig.3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ovig ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ovig
StepHypRef Expression
1 3simpa 1150 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝐴𝑅𝐵𝑆))
2 eleq1 2825 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
3 eleq1 2825 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
42, 3bi2anan9 639 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
543adant3 1134 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
6 ovig.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
75, 6anbi12d 634 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜓)))
8 ovig.2 . . . 4 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
9 moanimv 2620 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
108, 9mpbir 234 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
11 ovig.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
127, 10, 11ovigg 7354 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (((𝐴𝑅𝐵𝑆) ∧ 𝜓) → (𝐴𝐹𝐵) = 𝐶))
131, 12mpand 695 1 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  ∃*wmo 2537  (class class class)co 7213  {coprab 7214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217
This theorem is referenced by:  addsrpr  10689  mulsrpr  10690
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