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Theorem ov6g 6763
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 6618 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 eqid 2621 . . . . . 6 𝑆 = 𝑆
3 biidd 252 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑆 = 𝑆𝑆 = 𝑆))
43copsex2g 4928 . . . . . 6 ((𝐴𝐺𝐵𝐻) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) ↔ 𝑆 = 𝑆))
52, 4mpbiri 248 . . . . 5 ((𝐴𝐺𝐵𝐻) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
653adant3 1079 . . . 4 ((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
76adantr 481 . . 3 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
8 eqeq1 2625 . . . . . . . 8 (𝑤 = ⟨𝐴, 𝐵⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
98anbi1d 740 . . . . . . 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 2631 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝑧 = 𝑅𝑧 = 𝑆))
1211eqcoms 2629 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑧 = 𝑅𝑧 = 𝑆))
1312pm5.32i 668 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆))
149, 13syl6bb 276 . . . . . 6 (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
15142exbidv 1849 . . . . 5 (𝑤 = ⟨𝐴, 𝐵⟩ → (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
16 eqeq1 2625 . . . . . . 7 (𝑧 = 𝑆 → (𝑧 = 𝑆𝑆 = 𝑆))
1716anbi2d 739 . . . . . 6 (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
18172exbidv 1849 . . . . 5 (𝑧 = 𝑆 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
19 moeq 3369 . . . . . . 7 ∃*𝑧 𝑧 = 𝑅
2019mosubop 4943 . . . . . 6 ∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)
2120a1i 11 . . . . 5 (𝑤𝐶 → ∃*𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))
22 ov6g.2 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)}
23 dfoprab2 6666 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅))}
24 eleq1 2686 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐶 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
2524anbi1d 740 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤𝐶𝑧 = 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)))
2625pm5.32i 668 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤𝐶𝑧 = 𝑅)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)))
27 an12 837 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
2826, 27bitr3i 266 . . . . . . . . 9 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
29282exbii 1772 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ ∃𝑥𝑦(𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
30 19.42vv 1917 . . . . . . . 8 (∃𝑥𝑦(𝑤𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
3129, 30bitri 264 . . . . . . 7 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)) ↔ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
3231opabbii 4689 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
3322, 23, 323eqtri 2647 . . . . 5 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐶 ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
3415, 18, 21, 33fvopab3ig 6245 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ 𝐶𝑆𝐽) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
35343ad2antl3 1223 . . 3 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
367, 35mpd 15 . 2 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆)
371, 36syl5eq 2667 1 (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  ∃*wmo 2470  cop 4161  {copab 4682  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-fv 5865  df-ov 6618  df-oprab 6619
This theorem is referenced by: (None)
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