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Theorem ovid 6742
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 6618 . . 3 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
21eqeq1i 2626 . 2 ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3 ovid.1 . . . . . 6 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
43fnoprab 6728 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
5 ovid.2 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
65fneq1i 5953 . . . . 5 (𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
74, 6mpbir 221 . . . 4 𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
8 opabid 4952 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝑥𝑅𝑦𝑆))
98biimpri 218 . . . 4 ((𝑥𝑅𝑦𝑆) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
10 fnopfvb 6204 . . . 4 ((𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ∧ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
117, 9, 10sylancr 694 . . 3 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
125eleq2i 2690 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})
13 oprabid 6642 . . . . 5 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1412, 13bitri 264 . . . 4 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ((𝑥𝑅𝑦𝑆) ∧ 𝜑))
1514baib 943 . . 3 ((𝑥𝑅𝑦𝑆) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹𝜑))
1611, 15bitrd 268 . 2 ((𝑥𝑅𝑦𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝜑))
172, 16syl5bb 272 1 ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  ∃!weu 2469  cop 4161  {copab 4682   Fn wfn 5852  cfv 5857  (class class class)co 6615  {coprab 6616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-ov 6618  df-oprab 6619
This theorem is referenced by: (None)
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