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Theorem ov 6017
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ov.1 𝐶 ∈ V
ov.2 (𝑥 = 𝐴 → (𝜑𝜓))
ov.3 (𝑦 = 𝐵 → (𝜓𝜒))
ov.4 (𝑧 = 𝐶 → (𝜒𝜃))
ov.5 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
ov.6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ov ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶𝜃))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ov
StepHypRef Expression
1 df-ov 5900 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 ov.6 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
32fveq1i 5535 . . . . 5 (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
41, 3eqtri 2210 . . . 4 (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
54eqeq1i 2197 . . 3 ((𝐴𝐹𝐵) = 𝐶 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶)
6 ov.5 . . . . . 6 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
76fnoprab 6000 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
8 eleq1 2252 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
98anbi1d 465 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝑦𝑆)))
10 eleq1 2252 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
1110anbi2d 464 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
129, 11opelopabg 4286 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝐴𝑅𝐵𝑆)))
1312ibir 177 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
14 fnopfvb 5578 . . . . 5 (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
157, 13, 14sylancr 414 . . . 4 ((𝐴𝑅𝐵𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
16 ov.1 . . . . 5 𝐶 ∈ V
17 ov.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
189, 17anbi12d 473 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝑦𝑆) ∧ 𝜓)))
19 ov.3 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2011, 19anbi12d 473 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆) ∧ 𝜓) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜒)))
21 ov.4 . . . . . . 7 (𝑧 = 𝐶 → (𝜒𝜃))
2221anbi2d 464 . . . . . 6 (𝑧 = 𝐶 → (((𝐴𝑅𝐵𝑆) ∧ 𝜒) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2318, 20, 22eloprabg 5985 . . . . 5 ((𝐴𝑅𝐵𝑆𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2416, 23mp3an3 1337 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2515, 24bitrd 188 . . 3 ((𝐴𝑅𝐵𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
265, 25bitrid 192 . 2 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2726bianabs 611 1 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  ∃!weu 2038  wcel 2160  Vcvv 2752  cop 3610  {copab 4078   Fn wfn 5230  cfv 5235  (class class class)co 5897  {coprab 5898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5900  df-oprab 5901
This theorem is referenced by: (None)
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