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Theorem ov6g 5600
 Description: The value of an operation class abstraction. Special case. (Contributed by set.mm contributors, 13-Nov-2006.)
Hypotheses
Ref Expression
ov6g.1 (x, y = A, BR = 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 5526 . 2 (AFB) = (FA, B)
2 eqid 2353 . . . . . 6 S = S
3 biidd 228 . . . . . . 7 ((x = A y = B) → (S = SS = S))
43copsex2g 4609 . . . . . 6 ((A G B H) → (xy(A, B = x, y S = S) ↔ S = S))
52, 4mpbiri 224 . . . . 5 ((A G B H) → xy(A, B = x, y S = S))
653adant3 975 . . . 4 ((A G B H A, B C) → xy(A, B = x, y S = S))
76adantr 451 . . 3 (((A G B H A, B C) S J) → xy(A, B = x, y S = S))
8 eqeq1 2359 . . . . . . . 8 (w = A, B → (w = x, yA, B = x, y))
98anbi1d 685 . . . . . . 7 (w = A, B → ((w = x, y z = R) ↔ (A, B = x, y z = R)))
10 ov6g.1 . . . . . . . . . 10 (x, y = A, BR = S)
1110eqeq2d 2364 . . . . . . . . 9 (x, y = A, B → (z = Rz = S))
1211eqcoms 2356 . . . . . . . 8 (A, B = x, y → (z = Rz = S))
1312pm5.32i 618 . . . . . . 7 ((A, B = x, y z = R) ↔ (A, B = x, y z = S))
149, 13syl6bb 252 . . . . . 6 (w = A, B → ((w = x, y z = R) ↔ (A, B = x, y z = S)))
15142exbidv 1628 . . . . 5 (w = A, B → (xy(w = x, y z = R) ↔ xy(A, B = x, y z = S)))
16 eqeq1 2359 . . . . . . 7 (z = S → (z = SS = S))
1716anbi2d 684 . . . . . 6 (z = S → ((A, B = x, y z = S) ↔ (A, B = x, y S = S)))
18172exbidv 1628 . . . . 5 (z = S → (xy(A, B = x, y z = S) ↔ xy(A, B = x, y S = S)))
19 moeq 3012 . . . . . . 7 ∃*z z = R
2019mosubop 4613 . . . . . 6 ∃*zxy(w = x, y z = R)
2120a1i 10 . . . . 5 (w C∃*zxy(w = x, y z = R))
22 ov6g.2 . . . . . 6 F = {x, y, z (x, y C z = R)}
23 dfoprab2 5558 . . . . . 6 {x, y, z (x, y C z = R)} = {w, z xy(w = x, y (x, y C z = R))}
24 eleq1 2413 . . . . . . . . . . . 12 (w = x, y → (w Cx, y C))
2524anbi1d 685 . . . . . . . . . . 11 (w = x, y → ((w C z = R) ↔ (x, y C z = R)))
2625pm5.32i 618 . . . . . . . . . 10 ((w = x, y (w C z = R)) ↔ (w = x, y (x, y C z = R)))
27 an12 772 . . . . . . . . . 10 ((w = x, y (w C z = R)) ↔ (w C (w = x, y z = R)))
2826, 27bitr3i 242 . . . . . . . . 9 ((w = x, y (x, y C z = R)) ↔ (w C (w = x, y z = R)))
29282exbii 1583 . . . . . . . 8 (xy(w = x, y (x, y C z = R)) ↔ xy(w C (w = x, y z = R)))
30 19.42vv 1907 . . . . . . . 8 (xy(w C (w = x, y z = R)) ↔ (w C xy(w = x, y z = R)))
3129, 30bitri 240 . . . . . . 7 (xy(w = x, y (x, y C z = R)) ↔ (w C xy(w = x, y z = R)))
3231opabbii 4626 . . . . . 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 2377 . . . . 5 F = {w, z (w C xy(w = x, y z = R))}
3415, 18, 21, 33fvopab3ig 5387 . . . 4 ((A, B C S J) → (xy(A, B = x, y S = S) → (FA, B) = S))
35343ad2antl3 1119 . . 3 (((A G B H A, B C) S J) → (xy(A, B = x, y S = S) → (FA, B) = S))
367, 35mpd 14 . 2 (((A G B H A, B C) S J) → (FA, B) = S)
371, 36syl5eq 2397 1 (((A G B H A, B C) S J) → (AFB) = S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ⟨cop 4561  {copab 4622   ‘cfv 4781  (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-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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