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Theorem ttukeylem1 8136
Description: Lemma for ttukey 8145. 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 2796 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
21a1i 10 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  _V )
)
3 id 19 . . . . 5  |-  ( ( ~P C  i^i  Fin )  C_  A  ->  ( ~P C  i^i  Fin )  C_  A )
4 ssun1 3338 . . . . . . . 8  |-  U. A  C_  ( U. A  u.  B )
5 undif1 3529 . . . . . . . 8  |-  ( ( U. A  \  B
)  u.  B )  =  ( U. A  u.  B )
64, 5sseqtr4i 3211 . . . . . . 7  |-  U. A  C_  ( ( U. A  \  B )  u.  B
)
7 fvex 5539 . . . . . . . . 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 5479 . . . . . . . . . 10  |-  ( F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B )  ->  F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B ) )
108, 9syl 15 . . . . . . . . 9  |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B
) )
11 fornex 5750 . . . . . . . . 9  |-  ( (
card `  ( U. A  \  B ) )  e.  _V  ->  ( F : ( card `  ( U. A  \  B ) ) -onto-> ( U. A  \  B )  ->  ( U. A  \  B )  e.  _V ) )
127, 10, 11mpsyl 59 . . . . . . . 8  |-  ( ph  ->  ( U. A  \  B )  e.  _V )
13 ttukeylem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  A )
14 unexg 4521 . . . . . . . 8  |-  ( ( ( U. A  \  B )  e.  _V  /\  B  e.  A )  ->  ( ( U. A  \  B )  u.  B )  e.  _V )
1512, 13, 14syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( U. A  \  B )  u.  B
)  e.  _V )
16 ssexg 4160 . . . . . . 7  |-  ( ( U. A  C_  (
( U. A  \  B )  u.  B
)  /\  ( ( U. A  \  B )  u.  B )  e. 
_V )  ->  U. A  e.  _V )
176, 15, 16sylancr 644 . . . . . 6  |-  ( ph  ->  U. A  e.  _V )
18 uniexb 4563 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1917, 18sylibr 203 . . . . 5  |-  ( ph  ->  A  e.  _V )
20 ssexg 4160 . . . . 5  |-  ( ( ( ~P C  i^i  Fin )  C_  A  /\  A  e.  _V )  ->  ( ~P C  i^i  Fin )  e.  _V )
213, 19, 20syl2anr 464 . . . 4  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  ( ~P C  i^i  Fin )  e.  _V )
22 infpwfidom 7655 . . . 4  |-  ( ( ~P C  i^i  Fin )  e.  _V  ->  C  ~<_  ( ~P C  i^i  Fin ) )
23 reldom 6869 . . . . 5  |-  Rel  ~<_
2423brrelexi 4729 . . . 4  |-  ( C  ~<_  ( ~P C  i^i  Fin )  ->  C  e.  _V )
2521, 22, 243syl 18 . . 3  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  C  e.  _V )
2625ex 423 . 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 2343 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
29 pweq 3628 . . . . . . 7  |-  ( x  =  C  ->  ~P x  =  ~P C
)
3029ineq1d 3369 . . . . . 6  |-  ( x  =  C  ->  ( ~P x  i^i  Fin )  =  ( ~P C  i^i  Fin ) )
3130sseq1d 3205 . . . . 5  |-  ( x  =  C  ->  (
( ~P x  i^i 
Fin )  C_  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
3228, 31bibi12d 312 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A )  <->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A
) ) )
3332spcgv 2868 . . 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 26 . 2  |-  ( ph  ->  ( C  e.  _V  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) ) )
352, 26, 34pm5.21ndd 343 1  |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    ~<_ cdom 6861   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ttukeylem2  8137  ttukeylem6  8141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-en 6864  df-dom 6865  df-fin 6867
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