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Theorem ovi3 5560
Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovi3.1 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → 𝑆 (𝐻 × 𝐻))
ovi3.2 (((w = A v = B) (u = 𝐶 f = 𝐷)) → 𝑅 = 𝑆)
ovi3.3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
Assertion
Ref Expression
ovi3 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆)
Distinct variable groups:   u,f,v,w,x,y,z,A   B,f,u,v,w,x,y,z   x,𝑅,y,z   𝐶,f,u,v,w,y,z   𝐷,f,u,v,w,y,z   f,𝐻,u,v,w,x,y,z   𝑆,f,u,v,w,z
Allowed substitution hints:   𝐶(x)   𝐷(x)   𝑅(w,v,u,f)   𝑆(x,y)   𝐹(x,y,z,w,v,u,f)

Proof of Theorem ovi3
StepHypRef Expression
1 ovi3.1 . . . 4 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → 𝑆 (𝐻 × 𝐻))
2 elex 2543 . . . 4 (𝑆 (𝐻 × 𝐻) → 𝑆 V)
31, 2syl 14 . . 3 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → 𝑆 V)
4 isset 2539 . . 3 (𝑆 V ↔ z z = 𝑆)
53, 4sylib 127 . 2 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → z z = 𝑆)
6 nfv 1402 . . 3 z((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻))
7 nfcv 2160 . . . . 5 zA, B
8 ovi3.3 . . . . . 6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
9 nfoprab3 5479 . . . . . 6 z{⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
108, 9nfcxfr 2157 . . . . 5 z𝐹
11 nfcv 2160 . . . . 5 z𝐶, 𝐷
127, 10, 11nfov 5459 . . . 4 z(⟨A, B𝐹𝐶, 𝐷⟩)
1312nfeq1 2169 . . 3 z(⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆
14 ovi3.2 . . . . . . 7 (((w = A v = B) (u = 𝐶 f = 𝐷)) → 𝑅 = 𝑆)
1514eqeq2d 2033 . . . . . 6 (((w = A v = B) (u = 𝐶 f = 𝐷)) → (z = 𝑅z = 𝑆))
1615copsex4g 3958 . . . . 5 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅) ↔ z = 𝑆))
17 opelxpi 4303 . . . . . 6 ((A 𝐻 B 𝐻) → ⟨A, B (𝐻 × 𝐻))
18 opelxpi 4303 . . . . . 6 ((𝐶 𝐻 𝐷 𝐻) → ⟨𝐶, 𝐷 (𝐻 × 𝐻))
19 nfcv 2160 . . . . . . 7 xA, B
20 nfcv 2160 . . . . . . 7 yA, B
21 nfcv 2160 . . . . . . 7 y𝐶, 𝐷
22 nfv 1402 . . . . . . . 8 xwvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅)
23 nfoprab1 5477 . . . . . . . . . . 11 x{⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
248, 23nfcxfr 2157 . . . . . . . . . 10 x𝐹
25 nfcv 2160 . . . . . . . . . 10 xy
2619, 24, 25nfov 5459 . . . . . . . . 9 x(⟨A, B𝐹y)
2726nfeq1 2169 . . . . . . . 8 x(⟨A, B𝐹y) = z
2822, 27nfim 1446 . . . . . . 7 x(wvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹y) = z)
29 nfv 1402 . . . . . . . 8 ywvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅)
30 nfoprab2 5478 . . . . . . . . . . 11 y{⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
318, 30nfcxfr 2157 . . . . . . . . . 10 y𝐹
3220, 31, 21nfov 5459 . . . . . . . . 9 y(⟨A, B𝐹𝐶, 𝐷⟩)
3332nfeq1 2169 . . . . . . . 8 y(⟨A, B𝐹𝐶, 𝐷⟩) = z
3429, 33nfim 1446 . . . . . . 7 y(wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹𝐶, 𝐷⟩) = z)
35 eqeq1 2028 . . . . . . . . . . 11 (x = ⟨A, B⟩ → (x = ⟨w, v⟩ ↔ ⟨A, B⟩ = ⟨w, v⟩))
3635anbi1d 441 . . . . . . . . . 10 (x = ⟨A, B⟩ → ((x = ⟨w, v y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩)))
3736anbi1d 441 . . . . . . . . 9 (x = ⟨A, B⟩ → (((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ ((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅)))
38374exbidv 1732 . . . . . . . 8 (x = ⟨A, B⟩ → (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ wvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅)))
39 oveq1 5443 . . . . . . . . 9 (x = ⟨A, B⟩ → (x𝐹y) = (⟨A, B𝐹y))
4039eqeq1d 2030 . . . . . . . 8 (x = ⟨A, B⟩ → ((x𝐹y) = z ↔ (⟨A, B𝐹y) = z))
4138, 40imbi12d 223 . . . . . . 7 (x = ⟨A, B⟩ → ((wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) → (x𝐹y) = z) ↔ (wvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹y) = z)))
42 eqeq1 2028 . . . . . . . . . . 11 (y = ⟨𝐶, 𝐷⟩ → (y = ⟨u, f⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨u, f⟩))
4342anbi2d 440 . . . . . . . . . 10 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩)))
4443anbi1d 441 . . . . . . . . 9 (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ ((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅)))
45444exbidv 1732 . . . . . . . 8 (y = ⟨𝐶, 𝐷⟩ → (wvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅)))
46 oveq2 5444 . . . . . . . . 9 (y = ⟨𝐶, 𝐷⟩ → (⟨A, B𝐹y) = (⟨A, B𝐹𝐶, 𝐷⟩))
4746eqeq1d 2030 . . . . . . . 8 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B𝐹y) = z ↔ (⟨A, B𝐹𝐶, 𝐷⟩) = z))
4845, 47imbi12d 223 . . . . . . 7 (y = ⟨𝐶, 𝐷⟩ → ((wvuf((⟨A, B⟩ = ⟨w, v y = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹y) = z) ↔ (wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹𝐶, 𝐷⟩) = z)))
49 moeq 2693 . . . . . . . . . . . 12 ∃*z z = 𝑅
5049mosubop 4333 . . . . . . . . . . 11 ∃*zuf(y = ⟨u, f z = 𝑅)
5150mosubop 4333 . . . . . . . . . 10 ∃*zwv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅))
52 anass 383 . . . . . . . . . . . . . 14 (((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ (x = ⟨w, v (y = ⟨u, f z = 𝑅)))
53522exbii 1479 . . . . . . . . . . . . 13 (uf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ uf(x = ⟨w, v (y = ⟨u, f z = 𝑅)))
54 19.42vv 1770 . . . . . . . . . . . . 13 (uf(x = ⟨w, v (y = ⟨u, f z = 𝑅)) ↔ (x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
5553, 54bitri 173 . . . . . . . . . . . 12 (uf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ (x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
56552exbii 1479 . . . . . . . . . . 11 (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ wv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
5756mobii 1919 . . . . . . . . . 10 (∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ ∃*zwv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
5851, 57mpbir 134 . . . . . . . . 9 ∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅)
5958a1i 9 . . . . . . . 8 ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) → ∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))
6059, 8ovidi 5542 . . . . . . 7 ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) → (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) → (x𝐹y) = z))
6119, 20, 21, 28, 34, 41, 48, 60vtocl2gaf 2597 . . . . . 6 ((⟨A, B (𝐻 × 𝐻) 𝐶, 𝐷 (𝐻 × 𝐻)) → (wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹𝐶, 𝐷⟩) = z))
6217, 18, 61syl2an 273 . . . . 5 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (wvuf((⟨A, B⟩ = ⟨w, v𝐶, 𝐷⟩ = ⟨u, f⟩) z = 𝑅) → (⟨A, B𝐹𝐶, 𝐷⟩) = z))
6316, 62sylbird 159 . . . 4 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (z = 𝑆 → (⟨A, B𝐹𝐶, 𝐷⟩) = z))
64 eqeq2 2031 . . . 4 (z = 𝑆 → ((⟨A, B𝐹𝐶, 𝐷⟩) = z ↔ (⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆))
6563, 64mpbidi 140 . . 3 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (z = 𝑆 → (⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆))
666, 13, 65exlimd 1470 . 2 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (z z = 𝑆 → (⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆))
675, 66mpd 13 1 (((A 𝐻 B 𝐻) (𝐶 𝐻 𝐷 𝐻)) → (⟨A, B𝐹𝐶, 𝐷⟩) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374  ∃*wmo 1883  Vcvv 2535  cop 3353   × cxp 4270  (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-in1 532  ax-in2 533  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  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  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-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  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:  oviec  6123  addcnsr  6545  mulcnsr  6546
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