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Theorem ovigg 5857
Description: The value of an operation class abstraction. Compare ovig 5858. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovigg.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
ovigg.4 ∃*𝑧𝜑
ovigg.5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Assertion
Ref Expression
ovigg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem ovigg
StepHypRef Expression
1 ovigg.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
21eloprabga 5824 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
3 df-ov 5743 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 ovigg.5 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
54fveq1i 5388 . . . 4 (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2136 . . 3 (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
7 ovigg.4 . . . . 5 ∃*𝑧𝜑
87funoprab 5837 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
9 funopfv 5427 . . . 4 (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶))
108, 9ax-mp 5 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶)
116, 10syl5eq 2160 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶)
122, 11syl6bir 163 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 945   = wceq 1314  wcel 1463  ∃*wmo 1976  cop 3498  Fun wfun 5085  cfv 5091  (class class class)co 5740  {coprab 5741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744
This theorem is referenced by:  ovig  5858
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