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Theorem ov6g 5764
Description: The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
Hypotheses
Ref Expression
ov6g.1 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)
ov6g.2 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)}
Assertion
Ref Expression
ov6g (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑧,𝑅   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝐽(𝑥,𝑦,𝑧)

Proof of Theorem ov6g
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-ov 5637 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 eqid 2088 . . . . . 6 𝑆 = 𝑆
3 biidd 170 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑆 = 𝑆𝑆 = 𝑆))
43copsex2g 4064 . . . . . 6 ((𝐴𝐺𝐵𝐻) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) ↔ 𝑆 = 𝑆))
52, 4mpbiri 166 . . . . 5 ((𝐴𝐺𝐵𝐻) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
653adant3 963 . . . 4 ((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
76adantr 270 . . 3 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
8 eqeq1 2094 . . . . . . . 8 (𝑤 = ⟨𝐴, 𝐵⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
98anbi1d 453 . . . . . . 7 (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
10 ov6g.1 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)
1110eqeq2d 2099 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝑧 = 𝑅𝑧 = 𝑆))
1211eqcoms 2091 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑧 = 𝑅𝑧 = 𝑆))
1312pm5.32i 442 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆))
149, 13syl6bb 194 . . . . . 6 (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
15142exbidv 1796 . . . . 5 (𝑤 = ⟨𝐴, 𝐵⟩ → (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
16 eqeq1 2094 . . . . . . 7 (𝑧 = 𝑆 → (𝑧 = 𝑆𝑆 = 𝑆))
1716anbi2d 452 . . . . . 6 (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
18172exbidv 1796 . . . . 5 (𝑧 = 𝑆 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
19 moeq 2788 . . . . . . 7 ∃*𝑧 𝑧 = 𝑅
2019mosubop 4492 . . . . . 6 ∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)
2120a1i 9 . . . . 5 (𝑤𝐶 → ∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))
22 ov6g.2 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)}
23 dfoprab2 5678 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅))}
24 eleq1 2150 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐶 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
2524anbi1d 453 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤𝐶𝑧 = 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)))
2625pm5.32i 442 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤𝐶𝑧 = 𝑅)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)))
27 an12 528 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
2826, 27bitr3i 184 . . . . . . . . 9 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
29282exbii 1542 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ ∃𝑥𝑦(𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
30 19.42vv 1836 . . . . . . . 8 (∃𝑥𝑦(𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
3129, 30bitri 182 . . . . . . 7 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
3231opabbii 3897 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
3322, 23, 323eqtri 2112 . . . . 5 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
3415, 18, 21, 33fvopab3ig 5362 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ 𝐶𝑆𝐽) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
35343ad2antl3 1107 . . 3 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
367, 35mpd 13 . 2 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆)
371, 36syl5eq 2132 1 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wex 1426  wcel 1438  ∃*wmo 1949  cop 3444  {copab 3890  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-fv 5010  df-ov 5637  df-oprab 5638
This theorem is referenced by: (None)
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