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Theorem ovid 5743
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovid.1 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
ovid.2 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ovid ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
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 ovid
StepHypRef Expression
1 df-ov 5637 . . 3 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
21eqeq1i 2095 . 2 ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3 ovid.1 . . . . . 6 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
43fnoprab 5730 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
5 ovid.2 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
65fneq1i 5094 . . . . 5 (𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
74, 6mpbir 144 . . . 4 𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
8 opabid 4075 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝑥𝑅𝑦𝑆))
98biimpri 131 . . . 4 ((𝑥𝑅𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
10 fnopfvb 5330 . . . 4 ((𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ∧ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
117, 9, 10sylancr 405 . . 3 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
125eleq2i 2154 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})
13 oprabid 5663 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1412, 13bitri 182 . . . 4 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1514baib 866 . . 3 ((𝑥𝑅𝑦𝑆) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹𝜑))
1611, 15bitrd 186 . 2 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝜑))
172, 16syl5bb 190 1 ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  ∃!weu 1948  cop 3444  {copab 3890   Fn wfn 4997  cfv 5002  (class class class)co 5634  {coprab 5635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-ov 5637  df-oprab 5638
This theorem is referenced by: (None)
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