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Theorem ovid 7574
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 7434 . . 3 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
21eqeq1i 2742 . 2 ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3 ovid.1 . . . . . 6 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
43fnoprab 7558 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
5 ovid.2 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
65fneq1i 6665 . . . . 5 (𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
74, 6mpbir 231 . . . 4 𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
8 opabidw 5529 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝑥𝑅𝑦𝑆))
98biimpri 228 . . . 4 ((𝑥𝑅𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
10 fnopfvb 6960 . . . 4 ((𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ∧ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
117, 9, 10sylancr 587 . . 3 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
125eleq2i 2833 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})
13 oprabidw 7462 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1412, 13bitri 275 . . . 4 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1514baib 535 . . 3 ((𝑥𝑅𝑦𝑆) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹𝜑))
1611, 15bitrd 279 . 2 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝜑))
172, 16bitrid 283 1 ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  cop 4632  {copab 5205   Fn wfn 6556  cfv 6561  (class class class)co 7431  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435
This theorem is referenced by: (None)
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