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Theorem ovidig 6763
Description: The value of an operation class abstraction. Compare ovidi 6764. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovidig.1 ∃*𝑧𝜑
ovidig.2 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Assertion
Ref Expression
ovidig (𝜑 → (𝑥𝐹𝑦) = 𝑧)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ovidig
StepHypRef Expression
1 df-ov 6638 . 2 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2 ovidig.1 . . . . 5 ∃*𝑧𝜑
32funoprab 6745 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4 ovidig.2 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
54funeqi 5897 . . . 4 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
63, 5mpbir 221 . . 3 Fun 𝐹
7 oprabid 6662 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
87biimpri 218 . . . 4 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
98, 4syl6eleqr 2710 . . 3 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10 funopfv 6222 . . 3 (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
116, 9, 10mpsyl 68 . 2 (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
121, 11syl5eq 2666 1 (𝜑 → (𝑥𝐹𝑦) = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  ∃*wmo 2469  cop 4174  Fun wfun 5870  cfv 5876  (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:  ovidi  6764
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