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Theorem ttukeylem1 8152
Description: Lemma for ttukey 8161. 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 2809 . . 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 3351 . . . . . . . 8  |-  U. A  C_  ( U. A  u.  B )
5 undif1 3542 . . . . . . . 8  |-  ( ( U. A  \  B
)  u.  B )  =  ( U. A  u.  B )
64, 5sseqtr4i 3224 . . . . . . 7  |-  U. A  C_  ( ( U. A  \  B )  u.  B
)
7 fvex 5555 . . . . . . . . 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 5495 . . . . . . . . . 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 5766 . . . . . . . . 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 4537 . . . . . . . 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 4176 . . . . . . 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 4579 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1917, 18sylibr 203 . . . . 5  |-  ( ph  ->  A  e.  _V )
20 ssexg 4176 . . . . 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 7671 . . . 4  |-  ( ( ~P C  i^i  Fin )  e.  _V  ->  C  ~<_  ( ~P C  i^i  Fin ) )
23 reldom 6885 . . . . 5  |-  Rel  ~<_
2423brrelexi 4745 . . . 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 2356 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
29 pweq 3641 . . . . . . 7  |-  ( x  =  C  ->  ~P x  =  ~P C
)
3029ineq1d 3382 . . . . . 6  |-  ( x  =  C  ->  ( ~P x  i^i  Fin )  =  ( ~P C  i^i  Fin ) )
3130sseq1d 3218 . . . . 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 2881 . . 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 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    ~<_ cdom 6877   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  ttukeylem2  8153  ttukeylem6  8157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-en 6880  df-dom 6881  df-fin 6883
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