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Theorem ttukeylem1 8878
Description: Lemma for ttukey 8887. 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 3660 . . . . . . . 8  |-  U. A  C_  ( U. A  u.  B )
5 undif1 3895 . . . . . . . 8  |-  ( ( U. A  \  B
)  u.  B )  =  ( U. A  u.  B )
64, 5sseqtr4i 3530 . . . . . . 7  |-  U. A  C_  ( ( U. A  \  B )  u.  B
)
7 fvex 5867 . . . . . . . . 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 5814 . . . . . . . . . 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 6743 . . . . . . . . 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 6576 . . . . . . . 8  |-  ( ( ( U. A  \  B )  e.  _V  /\  B  e.  A )  ->  ( ( U. A  \  B )  u.  B )  e.  _V )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( U. A  \  B )  u.  B
)  e.  _V )
16 ssexg 4586 . . . . . . 7  |-  ( ( U. A  C_  (
( U. A  \  B )  u.  B
)  /\  ( ( U. A  \  B )  u.  B )  e. 
_V )  ->  U. A  e.  _V )
176, 15, 16sylancr 663 . . . . . 6  |-  ( ph  ->  U. A  e.  _V )
18 uniexb 6581 . . . . . 6  |-  ( A  e.  _V  <->  U. A  e. 
_V )
1917, 18sylibr 212 . . . . 5  |-  ( ph  ->  A  e.  _V )
20 ssexg 4586 . . . . 5  |-  ( ( ( ~P C  i^i  Fin )  C_  A  /\  A  e.  _V )  ->  ( ~P C  i^i  Fin )  e.  _V )
213, 19, 20syl2anr 478 . . . 4  |-  ( (
ph  /\  ( ~P C  i^i  Fin )  C_  A )  ->  ( ~P C  i^i  Fin )  e.  _V )
22 infpwfidom 8398 . . . 4  |-  ( ( ~P C  i^i  Fin )  e.  _V  ->  C  ~<_  ( ~P C  i^i  Fin ) )
23 reldom 7512 . . . . 5  |-  Rel  ~<_
2423brrelexi 5032 . . . 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 434 . 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 2532 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
29 pweq 4006 . . . . . . 7  |-  ( x  =  C  ->  ~P x  =  ~P C
)
3029ineq1d 3692 . . . . . 6  |-  ( x  =  C  ->  ( ~P x  i^i  Fin )  =  ( ~P C  i^i  Fin ) )
3130sseq1d 3524 . . . . 5  |-  ( x  =  C  ->  (
( ~P x  i^i 
Fin )  C_  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
3228, 31bibi12d 321 . . . 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 354 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 369   A.wal 1372    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   class class class wbr 4440   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579    ~<_ cdom 7504   Fincfn 7506   cardccrd 8305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-en 7507  df-dom 7508  df-fin 7510
This theorem is referenced by:  ttukeylem2  8879  ttukeylem6  8883
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