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Theorem ovig 6767
Description: The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (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 1056 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝐴𝑅𝐵𝑆))
2 eleq1 2687 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
3 eleq1 2687 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
42, 3bi2anan9 916 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
543adant3 1079 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
6 ovig.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
75, 6anbi12d 746 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜓)))
8 ovig.2 . . . 4 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
9 moanimv 2529 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
108, 9mpbir 221 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
11 ovig.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
127, 10, 11ovigg 6766 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (((𝐴𝑅𝐵𝑆) ∧ 𝜓) → (𝐴𝐹𝐵) = 𝐶))
131, 12mpand 710 1 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  ∃*wmo 2469  (class class class)co 6635  {coprab 6636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639
This theorem is referenced by:  addsrpr  9881  mulsrpr  9882
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