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Theorem pwfilem 7826
Description: Lemma for pwfi 7827. (Contributed by NM, 26-Mar-2007.)
Hypothesis
Ref Expression
pwfilem.1  |-  F  =  ( c  e.  ~P b  |->  ( c  u. 
{ x } ) )
Assertion
Ref Expression
pwfilem  |-  ( ~P b  e.  Fin  ->  ~P ( b  u.  {
x } )  e. 
Fin )
Distinct variable groups:    b, c    x, c
Allowed substitution hints:    F( x, b, c)

Proof of Theorem pwfilem
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 pwundif 4793 . 2  |-  ~P (
b  u.  { x } )  =  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)
2 vex 3121 . . . . . . . . 9  |-  c  e. 
_V
3 snex 4694 . . . . . . . . 9  |-  { x }  e.  _V
42, 3unex 6593 . . . . . . . 8  |-  ( c  u.  { x }
)  e.  _V
5 pwfilem.1 . . . . . . . 8  |-  F  =  ( c  e.  ~P b  |->  ( c  u. 
{ x } ) )
64, 5fnmpti 5715 . . . . . . 7  |-  F  Fn  ~P b
7 dffn4 5807 . . . . . . 7  |-  ( F  Fn  ~P b  <->  F : ~P b -onto-> ran  F )
86, 7mpbi 208 . . . . . 6  |-  F : ~P b -onto-> ran  F
9 fodomfi 7811 . . . . . 6  |-  ( ( ~P b  e.  Fin  /\  F : ~P b -onto-> ran  F )  ->  ran  F  ~<_  ~P b )
108, 9mpan2 671 . . . . 5  |-  ( ~P b  e.  Fin  ->  ran 
F  ~<_  ~P b )
11 domfi 7753 . . . . 5  |-  ( ( ~P b  e.  Fin  /\ 
ran  F  ~<_  ~P b
)  ->  ran  F  e. 
Fin )
1210, 11mpdan 668 . . . 4  |-  ( ~P b  e.  Fin  ->  ran 
F  e.  Fin )
13 eldifi 3631 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ~P ( b  u. 
{ x } ) )
143elpwun 6608 . . . . . . . . 9  |-  ( d  e.  ~P ( b  u.  { x }
)  <->  ( d  \  { x } )  e.  ~P b )
1513, 14sylib 196 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  \  { x } )  e.  ~P b )
16 undif1 3908 . . . . . . . . 9  |-  ( ( d  \  { x } )  u.  {
x } )  =  ( d  u.  {
x } )
17 elpwunsn 4074 . . . . . . . . . . 11  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  x  e.  d )
1817snssd 4178 . . . . . . . . . 10  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  { x }  C_  d )
19 ssequn2 3682 . . . . . . . . . 10  |-  ( { x }  C_  d  <->  ( d  u.  { x } )  =  d )
2018, 19sylib 196 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  u.  { x } )  =  d )
2116, 20syl5req 2521 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  =  ( ( d 
\  { x }
)  u.  { x } ) )
22 uneq1 3656 . . . . . . . . . 10  |-  ( c  =  ( d  \  { x } )  ->  ( c  u. 
{ x } )  =  ( ( d 
\  { x }
)  u.  { x } ) )
2322eqeq2d 2481 . . . . . . . . 9  |-  ( c  =  ( d  \  { x } )  ->  ( d  =  ( c  u.  {
x } )  <->  d  =  ( ( d  \  { x } )  u.  { x }
) ) )
2423rspcev 3219 . . . . . . . 8  |-  ( ( ( d  \  {
x } )  e. 
~P b  /\  d  =  ( ( d 
\  { x }
)  u.  { x } ) )  ->  E. c  e.  ~P  b d  =  ( c  u.  { x } ) )
2515, 21, 24syl2anc 661 . . . . . . 7  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
265, 4elrnmpti 5259 . . . . . . 7  |-  ( d  e.  ran  F  <->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
2725, 26sylibr 212 . . . . . 6  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ran  F )
2827ssriv 3513 . . . . 5  |-  ( ~P ( b  u.  {
x } )  \  ~P b )  C_  ran  F
29 ssdomg 7573 . . . . 5  |-  ( ran 
F  e.  Fin  ->  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  C_  ran  F  -> 
( ~P ( b  u.  { x }
)  \  ~P b
)  ~<_  ran  F )
)
3012, 28, 29mpisyl 18 . . . 4  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  ~<_  ran  F )
31 domfi 7753 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ~P ( b  u.  { x }
)  \  ~P b
)  ~<_  ran  F )  ->  ( ~P ( b  u.  { x }
)  \  ~P b
)  e.  Fin )
3212, 30, 31syl2anc 661 . . 3  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  e.  Fin )
33 unfi 7799 . . 3  |-  ( ( ( ~P ( b  u.  { x }
)  \  ~P b
)  e.  Fin  /\  ~P b  e.  Fin )  ->  ( ( ~P ( b  u.  {
x } )  \  ~P b )  u.  ~P b )  e.  Fin )
3432, 33mpancom 669 . 2  |-  ( ~P b  e.  Fin  ->  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)  e.  Fin )
351, 34syl5eqel 2559 1  |-  ( ~P b  e.  Fin  ->  ~P ( b  u.  {
x } )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   E.wrex 2818    \ cdif 3478    u. cun 3479    C_ wss 3481   ~Pcpw 4016   {csn 4033   class class class wbr 4453    |-> cmpt 4511   ran crn 5006    Fn wfn 5589   -onto->wfo 5592    ~<_ cdom 7526   Fincfn 7528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-fin 7532
This theorem is referenced by:  pwfi  7827
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