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Theorem ovidi 7285
 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 2698 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
31, 2mpbir 233 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
4 ovidi.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
53, 4ovidig 7284 . 2 (((𝑥𝑅𝑦𝑆) ∧ 𝜑) → (𝑥𝐹𝑦) = 𝑧)
65ex 415 1 ((𝑥𝑅𝑦𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∃*wmo 2614  (class class class)co 7148  {coprab 7149 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152 This theorem is referenced by:  ovmpt4g  7289  ov3  7303
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