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Theorem ov6g 5580
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 5458 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 eqid 2037 . . . . . 6 𝑆 = 𝑆
3 biidd 161 . . . . . . 7 ((x = A y = B) → (𝑆 = 𝑆𝑆 = 𝑆))
43copsex2g 3974 . . . . . 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 923 . . . 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 2043 . . . . . . . 8 (w = ⟨A, B⟩ → (w = ⟨x, y⟩ ↔ ⟨A, B⟩ = ⟨x, y⟩))
98anbi1d 438 . . . . . . 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 2048 . . . . . . . . 9 (⟨x, y⟩ = ⟨A, B⟩ → (z = 𝑅z = 𝑆))
1211eqcoms 2040 . . . . . . . 8 (⟨A, B⟩ = ⟨x, y⟩ → (z = 𝑅z = 𝑆))
1312pm5.32i 427 . . . . . . 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 1745 . . . . 5 (w = ⟨A, B⟩ → (xy(w = ⟨x, y z = 𝑅) ↔ xy(⟨A, B⟩ = ⟨x, y z = 𝑆)))
16 eqeq1 2043 . . . . . . 7 (z = 𝑆 → (z = 𝑆𝑆 = 𝑆))
1716anbi2d 437 . . . . . 6 (z = 𝑆 → ((⟨A, B⟩ = ⟨x, y z = 𝑆) ↔ (⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆)))
18172exbidv 1745 . . . . 5 (z = 𝑆 → (xy(⟨A, B⟩ = ⟨x, y z = 𝑆) ↔ xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆)))
19 moeq 2710 . . . . . . 7 ∃*z z = 𝑅
2019mosubop 4349 . . . . . 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 5494 . . . . . 6 {⟨⟨x, y⟩, z⟩ ∣ (⟨x, y 𝐶 z = 𝑅)} = {⟨w, z⟩ ∣ xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅))}
24 eleq1 2097 . . . . . . . . . . . 12 (w = ⟨x, y⟩ → (w 𝐶 ↔ ⟨x, y 𝐶))
2524anbi1d 438 . . . . . . . . . . 11 (w = ⟨x, y⟩ → ((w 𝐶 z = 𝑅) ↔ (⟨x, y 𝐶 z = 𝑅)))
2625pm5.32i 427 . . . . . . . . . 10 ((w = ⟨x, y (w 𝐶 z = 𝑅)) ↔ (w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)))
27 an12 495 . . . . . . . . . 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 1494 . . . . . . . 8 (xy(w = ⟨x, y (⟨x, y 𝐶 z = 𝑅)) ↔ xy(w 𝐶 (w = ⟨x, y z = 𝑅)))
30 19.42vv 1785 . . . . . . . 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 3815 . . . . . 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 2061 . . . . 5 𝐹 = {⟨w, z⟩ ∣ (w 𝐶 xy(w = ⟨x, y z = 𝑅))}
3415, 18, 21, 33fvopab3ig 5189 . . . 4 ((⟨A, B 𝐶 𝑆 𝐽) → (xy(⟨A, B⟩ = ⟨x, y 𝑆 = 𝑆) → (𝐹‘⟨A, B⟩) = 𝑆))
35343ad2antl3 1067 . . 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 2081 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 884   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  cop 3370  {copab 3808  cfv 4845  (class class class)co 5455  {coprab 5456
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459
This theorem is referenced by: (None)
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