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Theorem ov6g 5561
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 𝐶 z = 𝑅)}
Assertion
Ref Expression
ov6g (((A 𝐺 B 𝐻 A, B 𝐶) 𝑆 𝐽) → (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 5439 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 eqid 2022 . . . . . 6 𝑆 = 𝑆
3 biidd 161 . . . . . . 7 ((x = A y = B) → (𝑆 = 𝑆𝑆 = 𝑆))
43copsex2g 3957 . . . . . 6 ((A 𝐺 B 𝐻) → (xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆) ↔ 𝑆 = 𝑆))
52, 4mpbiri 157 . . . . 5 ((A 𝐺 B 𝐻) → xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆))
653adant3 912 . . . 4 ((A 𝐺 B 𝐻 A, B 𝐶) → xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆))
76adantr 261 . . 3 (((A 𝐺 B 𝐻 A, B 𝐶) 𝑆 𝐽) → xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆))
8 eqeq1 2028 . . . . . . . 8 (w = ⟨A, B⟩ → (w = ⟨x, y⟩ ↔ ⟨A, B⟩ = ⟨x, y⟩))
98anbi1d 441 . . . . . . 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 2033 . . . . . . . . 9 (⟨x, y⟩ = ⟨A, B⟩ → (z = 𝑅z = 𝑆))
1211eqcoms 2025 . . . . . . . 8 (⟨A, B⟩ = ⟨x, y⟩ → (z = 𝑅z = 𝑆))
1312pm5.32i 430 . . . . . . 7 ((⟨A, B⟩ = ⟨x, y z = 𝑅) ↔ (⟨A, B⟩ = ⟨x, y z = 𝑆))
149, 13syl6bb 185 . . . . . 6 (w = ⟨A, B⟩ → ((w = ⟨x, y z = 𝑅) ↔ (⟨A, B⟩ = ⟨x, y z = 𝑆)))
15142exbidv 1730 . . . . 5 (w = ⟨A, B⟩ → (xy(w = ⟨x, y z = 𝑅) ↔ xy(⟨A, B⟩ = ⟨x, y z = 𝑆)))
16 eqeq1 2028 . . . . . . 7 (z = 𝑆 → (z = 𝑆𝑆 = 𝑆))
1716anbi2d 440 . . . . . 6 (z = 𝑆 → ((⟨A, B⟩ = ⟨x, y z = 𝑆) ↔ (⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆)))
18172exbidv 1730 . . . . 5 (z = 𝑆 → (xy(⟨A, B⟩ = ⟨x, y z = 𝑆) ↔ xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆)))
19 moeq 2693 . . . . . . 7 ∃*z z = 𝑅
2019mosubop 4333 . . . . . 6 ∃*zxy(w = ⟨x, y z = 𝑅)
2120a1i 9 . . . . 5 (w 𝐶∃*zxy(w = ⟨x, y z = 𝑅))
22 ov6g.2 . . . . . 6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ (⟨x, y 𝐶 z = 𝑅)}
23 dfoprab2 5475 . . . . . 6 {⟨⟨x, y⟩, z⟩ ∣ (⟨x, y 𝐶 z = 𝑅)} = {⟨w, z⟩ ∣ xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅))}
24 eleq1 2082 . . . . . . . . . . . 12 (w = ⟨x, y⟩ → (w 𝐶 ↔ ⟨x, y 𝐶))
2524anbi1d 441 . . . . . . . . . . 11 (w = ⟨x, y⟩ → ((w 𝐶 z = 𝑅) ↔ (⟨x, y 𝐶 z = 𝑅)))
2625pm5.32i 430 . . . . . . . . . 10 ((w = ⟨x, y (w 𝐶 z = 𝑅)) ↔ (w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)))
27 an12 483 . . . . . . . . . 10 ((w = ⟨x, y (w 𝐶 z = 𝑅)) ↔ (w 𝐶 (w = ⟨x, y z = 𝑅)))
2826, 27bitr3i 175 . . . . . . . . 9 ((w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)) ↔ (w 𝐶 (w = ⟨x, y z = 𝑅)))
29282exbii 1479 . . . . . . . 8 (xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)) ↔ xy(w 𝐶 (w = ⟨x, y z = 𝑅)))
30 19.42vv 1770 . . . . . . . 8 (xy(w 𝐶 (w = ⟨x, y z = 𝑅)) ↔ (w 𝐶 xy(w = ⟨x, y z = 𝑅)))
3129, 30bitri 173 . . . . . . 7 (xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)) ↔ (w 𝐶 xy(w = ⟨x, y z = 𝑅)))
3231opabbii 3798 . . . . . 6 {⟨w, z⟩ ∣ xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅))} = {⟨w, z⟩ ∣ (w 𝐶 xy(w = ⟨x, y z = 𝑅))}
3322, 23, 323eqtri 2046 . . . . 5 𝐹 = {⟨w, z⟩ ∣ (w 𝐶 xy(w = ⟨x, y z = 𝑅))}
3415, 18, 21, 33fvopab3ig 5171 . . . 4 ((⟨A, B 𝐶 𝑆 𝐽) → (xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆) → (𝐹‘⟨A, B⟩) = 𝑆))
35343ad2antl3 1056 . . 3 (((A 𝐺 B 𝐻 A, B 𝐶) 𝑆 𝐽) → (xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆) → (𝐹‘⟨A, B⟩) = 𝑆))
367, 35mpd 13 . 2 (((A 𝐺 B 𝐻 A, B 𝐶) 𝑆 𝐽) → (𝐹‘⟨A, B⟩) = 𝑆)
371, 36syl5eq 2066 1 (((A 𝐺 B 𝐻 A, B 𝐶) 𝑆 𝐽) → (A𝐹B) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 873   = wceq 1228  wex 1362   wcel 1374  ∃*wmo 1883  cop 3353  {copab 3791  cfv 4829  (class class class)co 5436  {coprab 5437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440
This theorem is referenced by: (None)
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