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Theorem ovidig 7546
Description: The value of an operation class abstraction. Compare ovidi 7547. 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 7408 . 2 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2 ovidig.1 . . . . 5 ∃*𝑧𝜑
32funoprab 7526 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4 ovidig.2 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
54funeqi 6566 . . . 4 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
63, 5mpbir 230 . . 3 Fun 𝐹
7 oprabidw 7436 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
87biimpri 227 . . . 4 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
98, 4eleqtrrdi 2844 . . 3 (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10 funopfv 6940 . . 3 (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
116, 9, 10mpsyl 68 . 2 (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
121, 11eqtrid 2784 1 (𝜑 → (𝑥𝐹𝑦) = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  ∃*wmo 2532  cop 4633  Fun wfun 6534  cfv 6540  (class class class)co 7405  {coprab 7406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409
This theorem is referenced by:  ovidi  7547
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