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Theorem fnfullfunlem1 5856
 Description: Lemma for fnfullfun 5858. Binary relationship over part one of the full function definition. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
fnfullfunlem1 (A(( I F) ( ∼ I F))B ↔ (AFB x(AFxx = B)))
Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fnfullfunlem1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A(( I F) ( ∼ I F))B → (A V B V))
21simprd 449 . 2 (A(( I F) ( ∼ I F))BB V)
3 brex 4689 . . . 4 (AFB → (A V B V))
43simprd 449 . . 3 (AFBB V)
54adantr 451 . 2 ((AFB x(AFxx = B)) → B V)
6 breq2 4643 . . . 4 (y = B → (A(( I F) ( ∼ I F))yA(( I F) ( ∼ I F))B))
7 breq2 4643 . . . . 5 (y = B → (AFyAFB))
8 eqeq2 2362 . . . . . . 7 (y = B → (x = yx = B))
98imbi2d 307 . . . . . 6 (y = B → ((AFxx = y) ↔ (AFxx = B)))
109albidv 1625 . . . . 5 (y = B → (x(AFxx = y) ↔ x(AFxx = B)))
117, 10anbi12d 691 . . . 4 (y = B → ((AFy x(AFxx = y)) ↔ (AFB x(AFxx = B))))
126, 11bibi12d 312 . . 3 (y = B → ((A(( I F) ( ∼ I F))y ↔ (AFy x(AFxx = y))) ↔ (A(( I F) ( ∼ I F))B ↔ (AFB x(AFxx = B)))))
13 brdif 4694 . . . 4 (A(( I F) ( ∼ I F))y ↔ (A( I F)y ¬ A( ∼ I F)y))
14 coi2 5095 . . . . . 6 ( I F) = F
1514breqi 4645 . . . . 5 (A( I F)yAFy)
16 brco 4883 . . . . . . 7 (A( ∼ I F)yx(AFx x ∼ I y))
17 df-br 4640 . . . . . . . . . 10 (x ∼ I yx, y ∼ I )
18 vex 2862 . . . . . . . . . . . . 13 x V
19 vex 2862 . . . . . . . . . . . . 13 y V
2018, 19opex 4588 . . . . . . . . . . . 12 x, y V
2120elcompl 3225 . . . . . . . . . . 11 (x, y ∼ I ↔ ¬ x, y I )
22 df-br 4640 . . . . . . . . . . . 12 (x I yx, y I )
2319ideq 4870 . . . . . . . . . . . 12 (x I yx = y)
2422, 23bitr3i 242 . . . . . . . . . . 11 (x, y I ↔ x = y)
2521, 24xchbinx 301 . . . . . . . . . 10 (x, y ∼ I ↔ ¬ x = y)
2617, 25bitri 240 . . . . . . . . 9 (x ∼ I y ↔ ¬ x = y)
2726anbi2i 675 . . . . . . . 8 ((AFx x ∼ I y) ↔ (AFx ¬ x = y))
2827exbii 1582 . . . . . . 7 (x(AFx x ∼ I y) ↔ x(AFx ¬ x = y))
29 exanali 1585 . . . . . . 7 (x(AFx ¬ x = y) ↔ ¬ x(AFxx = y))
3016, 28, 293bitrri 263 . . . . . 6 x(AFxx = y) ↔ A( ∼ I F)y)
3130con1bii 321 . . . . 5 A( ∼ I F)yx(AFxx = y))
3215, 31anbi12i 678 . . . 4 ((A( I F)y ¬ A( ∼ I F)y) ↔ (AFy x(AFxx = y)))
3313, 32bitri 240 . . 3 (A(( I F) ( ∼ I F))y ↔ (AFy x(AFxx = y)))
3412, 33vtoclg 2914 . 2 (B V → (A(( I F) ( ∼ I F))B ↔ (AFB x(AFxx = B))))
352, 5, 34pm5.21nii 342 1 (A(( I F) ( ∼ I F))B ↔ (AFB x(AFxx = B)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721   I cid 4763 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-id 4767  df-cnv 4785 This theorem is referenced by:  fnfullfunlem2  5857  fvfullfunlem1  5861  fvfullfunlem2  5862
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