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Theorem fvopab4t 5385
 Description: Closed theorem form of fvopab4 5389. (Contributed by set.mm contributors, 21-Feb-2013.)
Assertion
Ref Expression
fvopab4t ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (FA) = C)
Distinct variable groups:   x,y,A   y,B   x,C,y   x,D,y
Allowed substitution hints:   B(x)   F(x,y)   V(x,y)

Proof of Theorem fvopab4t
StepHypRef Expression
1 elex 2867 . . 3 (C VC V)
21anim2i 552 . 2 ((A D C V) → (A D C V))
3 funopab4 5141 . . . 4 Fun {x, y (x D y = B)}
4 simp2 956 . . . . . 6 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → x F = {x, y (x D y = B)})
5419.21bi 1758 . . . . 5 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → F = {x, y (x D y = B)})
65funeqd 5129 . . . 4 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (Fun F ↔ Fun {x, y (x D y = B)}))
73, 6mpbiri 224 . . 3 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → Fun F)
8 simp3l 983 . . . . 5 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → A D)
9 eqidd 2354 . . . . 5 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → C = C)
10 eleq1 2413 . . . . . . . . . . 11 (x = A → (x DA D))
11 eqeq2 2362 . . . . . . . . . . 11 (B = C → (y = By = C))
1210, 11bi2anan9 843 . . . . . . . . . 10 ((x = A B = C) → ((x D y = B) ↔ (A D y = C)))
1312ex 423 . . . . . . . . 9 (x = A → (B = C → ((x D y = B) ↔ (A D y = C))))
1413a2i 12 . . . . . . . 8 ((x = AB = C) → (x = A → ((x D y = B) ↔ (A D y = C))))
15142alimi 1560 . . . . . . 7 (xy(x = AB = C) → xy(x = A → ((x D y = B) ↔ (A D y = C))))
16153ad2ant1 976 . . . . . 6 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → xy(x = A → ((x D y = B) ↔ (A D y = C))))
17 eqeq1 2359 . . . . . . . . 9 (y = C → (y = CC = C))
1817anbi2d 684 . . . . . . . 8 (y = C → ((A D y = C) ↔ (A D C = C)))
1918gen2 1547 . . . . . . 7 xy(y = C → ((A D y = C) ↔ (A D C = C)))
2019a1i 10 . . . . . 6 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → xy(y = C → ((A D y = C) ↔ (A D C = C))))
21 simp3 957 . . . . . 6 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (A D C V))
22 opelopabt 4699 . . . . . 6 ((xy(x = A → ((x D y = B) ↔ (A D y = C))) xy(y = C → ((A D y = C) ↔ (A D C = C))) (A D C V)) → (A, C {x, y (x D y = B)} ↔ (A D C = C)))
2316, 20, 21, 22syl3anc 1182 . . . . 5 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (A, C {x, y (x D y = B)} ↔ (A D C = C)))
248, 9, 23mpbir2and 888 . . . 4 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → A, C {x, y (x D y = B)})
2524, 5eleqtrrd 2430 . . 3 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → A, C F)
26 funopfv 5357 . . 3 (Fun F → (A, C F → (FA) = C))
277, 25, 26sylc 56 . 2 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (FA) = C)
282, 27syl3an3 1217 1 ((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (FA) = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561  {copab 4622  Fun wfun 4775   ‘cfv 4781 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 This theorem is referenced by: (None)
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