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Theorem ovigg 7578
Description: The value of an operation class abstraction. Compared with ovig 7579, the condition (𝑥𝑅𝑦𝑆) is removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovigg.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
ovigg.4 ∃*𝑧𝜑
ovigg.5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Assertion
Ref Expression
ovigg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem ovigg
StepHypRef Expression
1 ovigg.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
21eloprabga 7542 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
3 df-ov 7434 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 ovigg.5 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
54fveq1i 6907 . . . 4 (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2765 . . 3 (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
7 ovigg.4 . . . . 5 ∃*𝑧𝜑
87funoprab 7555 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
9 funopfv 6958 . . . 4 (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶))
108, 9ax-mp 5 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶)
116, 10eqtrid 2789 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶)
122, 11biimtrrdi 254 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  ∃*wmo 2538  cop 4632  Fun wfun 6555  cfv 6561  (class class class)co 7431  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435
This theorem is referenced by:  ovig  7579  joinval  18422  meetval  18436
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