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Theorem ov3 5599
 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
ov3.1 S V
ov3.2 (((w = A v = B) (u = C f = D)) → R = S)
ov3.3 F = {x, y, z ((x (H × H) y (H × H)) wvuf((x = w, v y = u, f) z = R))}
Assertion
Ref Expression
ov3 (((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 ov3
StepHypRef Expression
1 ov3.1 . . 3 S V
21isseti 2865 . 2 z z = S
3 nfv 1619 . . 3 z((A H B H) (C H D H))
4 nfcv 2489 . . . . 5 zA, B
5 ov3.3 . . . . . 6 F = {x, y, z ((x (H × H) y (H × H)) wvuf((x = w, v y = u, f) z = R))}
6 nfoprab3 5548 . . . . . 6 z{x, y, z ((x (H × H) y (H × H)) wvuf((x = w, v y = u, f) z = R))}
75, 6nfcxfr 2486 . . . . 5 zF
8 nfcv 2489 . . . . 5 zC, D
94, 7, 8nfov 5545 . . . 4 z(A, BFC, D)
109nfeq1 2498 . . 3 z(A, BFC, D) = S
11 ov3.2 . . . . . . 7 (((w = A v = B) (u = C f = D)) → R = S)
1211eqeq2d 2364 . . . . . 6 (((w = A v = B) (u = C f = D)) → (z = Rz = S))
1312copsex4g 4610 . . . . 5 (((A H B H) (C H D H)) → (wvuf((A, B = w, v C, D = u, f) z = R) ↔ z = S))
14 opelxp 4811 . . . . . 6 (A, B (H × H) ↔ (A H B H))
15 opelxp 4811 . . . . . 6 (C, D (H × H) ↔ (C H D H))
16 nfcv 2489 . . . . . . 7 xA, B
17 nfcv 2489 . . . . . . 7 yA, B
18 nfcv 2489 . . . . . . 7 yC, D
19 nfv 1619 . . . . . . . 8 xwvuf((A, B = w, v y = u, f) z = R)
20 nfoprab1 5546 . . . . . . . . . . 11 x{x, y, z ((x (H × H) y (H × H)) wvuf((x = w, v y = u, f) z = R))}
215, 20nfcxfr 2486 . . . . . . . . . 10 xF
22 nfcv 2489 . . . . . . . . . 10 xy
2316, 21, 22nfov 5545 . . . . . . . . 9 x(A, BFy)
2423nfeq1 2498 . . . . . . . 8 x(A, BFy) = z
2519, 24nfim 1813 . . . . . . 7 x(wvuf((A, B = w, v y = u, f) z = R) → (A, BFy) = z)
26 nfv 1619 . . . . . . . 8 ywvuf((A, B = w, v C, D = u, f) z = R)
27 nfoprab2 5547 . . . . . . . . . . 11 y{x, y, z ((x (H × H) y (H × H)) wvuf((x = w, v y = u, f) z = R))}
285, 27nfcxfr 2486 . . . . . . . . . 10 yF
2917, 28, 18nfov 5545 . . . . . . . . 9 y(A, BFC, D)
3029nfeq1 2498 . . . . . . . 8 y(A, BFC, D) = z
3126, 30nfim 1813 . . . . . . 7 y(wvuf((A, B = w, v C, D = u, f) z = R) → (A, BFC, D) = z)
32 eqeq1 2359 . . . . . . . . . . 11 (x = A, B → (x = w, vA, B = w, v))
3332anbi1d 685 . . . . . . . . . 10 (x = A, B → ((x = w, v y = u, f) ↔ (A, B = w, v y = u, f)))
3433anbi1d 685 . . . . . . . . 9 (x = A, B → (((x = w, v y = u, f) z = R) ↔ ((A, B = w, v y = u, f) z = R)))
35344exbidv 1630 . . . . . . . 8 (x = A, B → (wvuf((x = w, v y = u, f) z = R) ↔ wvuf((A, B = w, v y = u, f) z = R)))
36 oveq1 5530 . . . . . . . . 9 (x = A, B → (xFy) = (A, BFy))
3736eqeq1d 2361 . . . . . . . 8 (x = A, B → ((xFy) = z ↔ (A, BFy) = z))
3835, 37imbi12d 311 . . . . . . 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)))
39 eqeq1 2359 . . . . . . . . . . 11 (y = C, D → (y = u, fC, D = u, f))
4039anbi2d 684 . . . . . . . . . 10 (y = C, D → ((A, B = w, v y = u, f) ↔ (A, B = w, v C, D = u, f)))
4140anbi1d 685 . . . . . . . . 9 (y = C, D → (((A, B = w, v y = u, f) z = R) ↔ ((A, B = w, v C, D = u, f) z = R)))
42414exbidv 1630 . . . . . . . 8 (y = C, D → (wvuf((A, B = w, v y = u, f) z = R) ↔ wvuf((A, B = w, v C, D = u, f) z = R)))
43 oveq2 5531 . . . . . . . . 9 (y = C, D → (A, BFy) = (A, BFC, D))
4443eqeq1d 2361 . . . . . . . 8 (y = C, D → ((A, BFy) = z ↔ (A, BFC, D) = z))
4542, 44imbi12d 311 . . . . . . 7 (y = C, D → ((wvuf((A, B = w, v y = u, f) z = R) → (A, BFy) = z) ↔ (wvuf((A, B = w, v C, D = u, f) z = R) → (A, BFC, D) = z)))
46 moeq 3012 . . . . . . . . . . . 12 ∃*z z = R
4746mosubop 4613 . . . . . . . . . . 11 ∃*zuf(y = u, f z = R)
4847mosubop 4613 . . . . . . . . . 10 ∃*zwv(x = w, v uf(y = u, f z = R))
49 anass 630 . . . . . . . . . . . . . 14 (((x = w, v y = u, f) z = R) ↔ (x = w, v (y = u, f z = R)))
50492exbii 1583 . . . . . . . . . . . . 13 (uf((x = w, v y = u, f) z = R) ↔ uf(x = w, v (y = u, f z = R)))
51 19.42vv 1907 . . . . . . . . . . . . 13 (uf(x = w, v (y = u, f z = R)) ↔ (x = w, v uf(y = u, f z = R)))
5250, 51bitri 240 . . . . . . . . . . . 12 (uf((x = w, v y = u, f) z = R) ↔ (x = w, v uf(y = u, f z = R)))
53522exbii 1583 . . . . . . . . . . 11 (wvuf((x = w, v y = u, f) z = R) ↔ wv(x = w, v uf(y = u, f z = R)))
5453mobii 2240 . . . . . . . . . 10 (∃*zwvuf((x = w, v y = u, f) z = R) ↔ ∃*zwv(x = w, v uf(y = u, f z = R)))
5548, 54mpbir 200 . . . . . . . . 9 ∃*zwvuf((x = w, v y = u, f) z = R)
5655a1i 10 . . . . . . . 8 ((x (H × H) y (H × H)) → ∃*zwvuf((x = w, v y = u, f) z = R))
5756, 5ovidi 5594 . . . . . . 7 ((x (H × H) y (H × H)) → (wvuf((x = w, v y = u, f) z = R) → (xFy) = z))
5816, 17, 18, 25, 31, 38, 45, 57vtocl2gaf 2921 . . . . . 6 ((A, B (H × H) C, D (H × H)) → (wvuf((A, B = w, v C, D = u, f) z = R) → (A, BFC, D) = z))
5914, 15, 58syl2anbr 466 . . . . 5 (((A H B H) (C H D H)) → (wvuf((A, B = w, v C, D = u, f) z = R) → (A, BFC, D) = z))
6013, 59sylbird 226 . . . 4 (((A H B H) (C H D H)) → (z = S → (A, BFC, D) = z))
61 eqeq2 2362 . . . 4 (z = S → ((A, BFC, D) = z ↔ (A, BFC, D) = S))
6260, 61mpbidi 207 . . 3 (((A H B H) (C H D H)) → (z = S → (A, BFC, D) = S))
633, 10, 62exlimd 1806 . 2 (((A H B H) (C H D H)) → (z z = S → (A, BFC, D) = S))
642, 63mpi 16 1 (((A H B H) (C H D H)) → (A, BFC, D) = S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2859  ⟨cop 4561   × cxp 4770  (class class class)co 5525  {coprab 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528 This theorem is referenced by: (None)
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