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Theorem ovig 5974
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 989 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝐴𝑅𝐵𝑆))
2 eleq1 2233 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
3 eleq1 2233 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
42, 3bi2anan9 601 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
543adant3 1012 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
6 ovig.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
75, 6anbi12d 470 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜓)))
8 ovig.2 . . . 4 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
9 moanimv 2094 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
108, 9mpbir 145 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
11 ovig.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
127, 10, 11ovigg 5973 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (((𝐴𝑅𝐵𝑆) ∧ 𝜓) → (𝐴𝐹𝐵) = 𝐶))
131, 12mpand 427 1 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  ∃*wmo 2020  wcel 2141  (class class class)co 5853  {coprab 5854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857
This theorem is referenced by:  th3q  6618  addnnnq0  7411  mulnnnq0  7412  addsrpr  7707  mulsrpr  7708
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