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Theorem ttukeylem1 8880
Description: Lemma for ttukey 8889. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )
ttukeylem.2  |-  ( ph  ->  B  e.  A )
ttukeylem.3  |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )
Assertion
Ref Expression
ttukeylem1  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
Distinct variable groups:    x, C    x, A    x, B    x, F
Allowed substitution hint:    ph( x)

Proof of Theorem ttukeylem1
StepHypRef Expression
1 elex 3115 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
21a1i 11 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  _V )
)
3 id 22 . . . . 5  |-  ( ( ~P C  i^i  Fin )  C_  A  ->  ( ~P C  i^i  Fin )  C_  A )
4 ssun1 3653 . . . . . . . 8  |-  U. A  C_  ( U. A  u.  B )
5 undif1 3891 . . . . . . . 8  |-  ( ( U. A  \  B
)  u.  B )  =  ( U. A  u.  B )
64, 5sseqtr4i 3522 . . . . . . 7  |-  U. A  C_  ( ( U. A  \  B )  u.  B
)
7 fvex 5858 . . . . . . . . 9  |-  ( card `  ( U. A  \  B ) )  e. 
_V
8 ttukeylem.1 . . . . . . . . . 10  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )
9 f1ofo 5805 . . . . . . . . . 10  |-  ( F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B )  ->  F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B ) )
108, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B
) )
11 fornex 6742 . . . . . . . . 9  |-  ( (
card `  ( U. A  \  B ) )  e.  _V  ->  ( F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B )  ->  ( U. A  \  B )  e.  _V ) )
127, 10, 11mpsyl 63 . . . . . . . 8  |-  ( ph  ->  ( U. A  \  B )  e.  _V )
13 ttukeylem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  A )
14 unexg 6574 . . . . . . . 8  |-  ( ( ( U. A  \  B )  e.  _V  /\  B  e.  A )  ->  ( ( U. A  \  B )  u.  B )  e.  _V )
1512, 13, 14syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( U. A  \  B )  u.  B
)  e.  _V )
16 ssexg 4583 . . . . . . 7  |-  ( ( U. A  C_  (
( U. A  \  B )  u.  B
)  /\  ( ( U. A  \  B )  u.  B )  e. 
_V )  ->  U. A  e.  _V )
176, 15, 16sylancr 661 . . . . . 6  |-  ( ph  ->  U. A  e.  _V )
18 uniexb 6583 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1917, 18sylibr 212 . . . . 5  |-  ( ph  ->  A  e.  _V )
20 ssexg 4583 . . . . 5  |-  ( ( ( ~P C  i^i  Fin )  C_  A  /\  A  e.  _V )  ->  ( ~P C  i^i  Fin )  e.  _V )
213, 19, 20syl2anr 476 . . . 4  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  ( ~P C  i^i  Fin )  e.  _V )
22 infpwfidom 8400 . . . 4  |-  ( ( ~P C  i^i  Fin )  e.  _V  ->  C  ~<_  ( ~P C  i^i  Fin ) )
23 reldom 7515 . . . . 5  |-  Rel  ~<_
2423brrelexi 5029 . . . 4  |-  ( C  ~<_  ( ~P C  i^i  Fin )  ->  C  e.  _V )
2521, 22, 243syl 20 . . 3  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  C  e.  _V )
2625ex 432 . 2  |-  ( ph  ->  ( ( ~P C  i^i  Fin )  C_  A  ->  C  e.  _V )
)
27 ttukeylem.3 . . 3  |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )
28 eleq1 2526 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
29 pweq 4002 . . . . . . 7  |-  ( x  =  C  ->  ~P x  =  ~P C
)
3029ineq1d 3685 . . . . . 6  |-  ( x  =  C  ->  ( ~P x  i^i  Fin )  =  ( ~P C  i^i  Fin ) )
3130sseq1d 3516 . . . . 5  |-  ( x  =  C  ->  (
( ~P x  i^i 
Fin )  C_  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
3228, 31bibi12d 319 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A )  <->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) ) )
3332spcgv 3191 . . 3  |-  ( C  e.  _V  ->  ( A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A
)  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) ) )
3427, 33syl5com 30 . 2  |-  ( ph  ->  ( C  e.  _V  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) ) )
352, 26, 34pm5.21ndd 352 1  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570    ~<_ cdom 7507   Fincfn 7509   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-en 7510  df-dom 7511  df-fin 7513
This theorem is referenced by:  ttukeylem2  8881  ttukeylem6  8885
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