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Theorem ov 5746
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 5637 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 ov.6 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
32fveq1i 5290 . . . . 5 (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
41, 3eqtri 2108 . . . 4 (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
54eqeq1i 2095 . . 3 ((𝐴𝐹𝐵) = 𝐶 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶)
6 ov.5 . . . . . 6 ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)
76fnoprab 5730 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}
8 eleq1 2150 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
98anbi1d 453 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝑦𝑆)))
10 eleq1 2150 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
1110anbi2d 452 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
129, 11opelopabg 4086 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝐴𝑅𝐵𝑆)))
1312ibir 175 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
14 fnopfvb 5330 . . . . 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 405 . . . 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 457 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝑦𝑆) ∧ 𝜓)))
19 ov.3 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
2011, 19anbi12d 457 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆) ∧ 𝜓) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜒)))
21 ov.4 . . . . . . 7 (𝑧 = 𝐶 → (𝜒𝜃))
2221anbi2d 452 . . . . . 6 (𝑧 = 𝐶 → (((𝐴𝑅𝐵𝑆) ∧ 𝜒) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2318, 20, 22eloprabg 5718 . . . . 5 ((𝐴𝑅𝐵𝑆𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2416, 23mp3an3 1262 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2515, 24bitrd 186 . . 3 ((𝐴𝑅𝐵𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
265, 25syl5bb 190 . 2 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
2726bianabs 578 1 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  ∃!weu 1948  Vcvv 2619  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-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
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  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-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  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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