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Theorem ovidig 6654
Description: The value of an operation class abstraction. Compare ovidi 6655. 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 6530 . 2 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2 ovidig.1 . . . . 5 ∃*𝑧𝜑
32funoprab 6636 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4 ovidig.2 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
54funeqi 5810 . . . 4 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
63, 5mpbir 219 . . 3 Fun 𝐹
7 oprabid 6554 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
87biimpri 216 . . . 4 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
98, 4syl6eleqr 2698 . . 3 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10 funopfv 6130 . . 3 (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
116, 9, 10mpsyl 65 . 2 (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
121, 11syl5eq 2655 1 (𝜑 → (𝑥𝐹𝑦) = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  ∃*wmo 2458  cop 4130  Fun wfun 5784  cfv 5790  (class class class)co 6527  {coprab 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531
This theorem is referenced by:  ovidi  6655
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