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Theorem ovidi 7564
Description: The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovidi.2 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
ovidi.3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ovidi ((𝑥𝑅𝑦𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝑅   𝑧,𝑆
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ovidi
StepHypRef Expression
1 ovidi.2 . . . 4 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
2 moanimv 2611 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
31, 2mpbir 230 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
4 ovidi.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
53, 4ovidig 7563 . 2 (((𝑥𝑅𝑦𝑆) ∧ 𝜑) → (𝑥𝐹𝑦) = 𝑧)
65ex 412 1 ((𝑥𝑅𝑦𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  ∃*wmo 2528  (class class class)co 7420  {coprab 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424
This theorem is referenced by:  ovmpt4g  7568  ov3  7584
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