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Theorem ttukeylem1 8381
Description: Lemma for ttukey 8390. 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 2956 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
21a1i 11 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  _V )
)
3 id 20 . . . . 5  |-  ( ( ~P C  i^i  Fin )  C_  A  ->  ( ~P C  i^i  Fin )  C_  A )
4 ssun1 3502 . . . . . . . 8  |-  U. A  C_  ( U. A  u.  B )
5 undif1 3695 . . . . . . . 8  |-  ( ( U. A  \  B
)  u.  B )  =  ( U. A  u.  B )
64, 5sseqtr4i 3373 . . . . . . 7  |-  U. A  C_  ( ( U. A  \  B )  u.  B
)
7 fvex 5734 . . . . . . . . 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 5673 . . . . . . . . . 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 5962 . . . . . . . . 9  |-  ( (
card `  ( U. A  \  B ) )  e.  _V  ->  ( F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B )  ->  ( U. A  \  B )  e.  _V ) )
127, 10, 11mpsyl 61 . . . . . . . 8  |-  ( ph  ->  ( U. A  \  B )  e.  _V )
13 ttukeylem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  A )
14 unexg 4702 . . . . . . . 8  |-  ( ( ( U. A  \  B )  e.  _V  /\  B  e.  A )  ->  ( ( U. A  \  B )  u.  B )  e.  _V )
1512, 13, 14syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( U. A  \  B )  u.  B
)  e.  _V )
16 ssexg 4341 . . . . . . 7  |-  ( ( U. A  C_  (
( U. A  \  B )  u.  B
)  /\  ( ( U. A  \  B )  u.  B )  e. 
_V )  ->  U. A  e.  _V )
176, 15, 16sylancr 645 . . . . . 6  |-  ( ph  ->  U. A  e.  _V )
18 uniexb 4744 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1917, 18sylibr 204 . . . . 5  |-  ( ph  ->  A  e.  _V )
20 ssexg 4341 . . . . 5  |-  ( ( ( ~P C  i^i  Fin )  C_  A  /\  A  e.  _V )  ->  ( ~P C  i^i  Fin )  e.  _V )
213, 19, 20syl2anr 465 . . . 4  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  ( ~P C  i^i  Fin )  e.  _V )
22 infpwfidom 7901 . . . 4  |-  ( ( ~P C  i^i  Fin )  e.  _V  ->  C  ~<_  ( ~P C  i^i  Fin ) )
23 reldom 7107 . . . . 5  |-  Rel  ~<_
2423brrelexi 4910 . . . 4  |-  ( C  ~<_  ( ~P C  i^i  Fin )  ->  C  e.  _V )
2521, 22, 243syl 19 . . 3  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  C  e.  _V )
2625ex 424 . 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 2495 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
29 pweq 3794 . . . . . . 7  |-  ( x  =  C  ->  ~P x  =  ~P C
)
3029ineq1d 3533 . . . . . 6  |-  ( x  =  C  ->  ( ~P x  i^i  Fin )  =  ( ~P C  i^i  Fin ) )
3130sseq1d 3367 . . . . 5  |-  ( x  =  C  ->  (
( ~P x  i^i 
Fin )  C_  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
3228, 31bibi12d 313 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A )  <->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) ) )
3332spcgv 3028 . . 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 28 . 2  |-  ( ph  ->  ( C  e.  _V  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) ) )
352, 26, 34pm5.21ndd 344 1  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   class class class wbr 4204   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    ~<_ cdom 7099   Fincfn 7101   cardccrd 7814
This theorem is referenced by:  ttukeylem2  8382  ttukeylem6  8386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-en 7102  df-dom 7103  df-fin 7105
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