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Theorem pwfilem 7886
Description: Lemma for pwfi 7887. (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 4746 . 2  |-  ~P (
b  u.  { x } )  =  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)
2 vex 3034 . . . . . . . . 9  |-  c  e. 
_V
3 snex 4641 . . . . . . . . 9  |-  { x }  e.  _V
42, 3unex 6608 . . . . . . . 8  |-  ( c  u.  { x }
)  e.  _V
5 pwfilem.1 . . . . . . . 8  |-  F  =  ( c  e.  ~P b  |->  ( c  u. 
{ x } ) )
64, 5fnmpti 5716 . . . . . . 7  |-  F  Fn  ~P b
7 dffn4 5812 . . . . . . 7  |-  ( F  Fn  ~P b  <->  F : ~P b -onto-> ran  F )
86, 7mpbi 213 . . . . . 6  |-  F : ~P b -onto-> ran  F
9 fodomfi 7868 . . . . . 6  |-  ( ( ~P b  e.  Fin  /\  F : ~P b -onto-> ran  F )  ->  ran  F  ~<_  ~P b )
108, 9mpan2 685 . . . . 5  |-  ( ~P b  e.  Fin  ->  ran 
F  ~<_  ~P b )
11 domfi 7811 . . . . 5  |-  ( ( ~P b  e.  Fin  /\ 
ran  F  ~<_  ~P b
)  ->  ran  F  e. 
Fin )
1210, 11mpdan 681 . . . 4  |-  ( ~P b  e.  Fin  ->  ran 
F  e.  Fin )
13 eldifi 3544 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ~P ( b  u. 
{ x } ) )
143elpwun 6623 . . . . . . . . 9  |-  ( d  e.  ~P ( b  u.  { x }
)  <->  ( d  \  { x } )  e.  ~P b )
1513, 14sylib 201 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  \  { x } )  e.  ~P b )
16 undif1 3833 . . . . . . . . 9  |-  ( ( d  \  { x } )  u.  {
x } )  =  ( d  u.  {
x } )
17 elpwunsn 4003 . . . . . . . . . . 11  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  x  e.  d )
1817snssd 4108 . . . . . . . . . 10  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  { x }  C_  d )
19 ssequn2 3598 . . . . . . . . . 10  |-  ( { x }  C_  d  <->  ( d  u.  { x } )  =  d )
2018, 19sylib 201 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  u.  { x } )  =  d )
2116, 20syl5req 2518 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  =  ( ( d 
\  { x }
)  u.  { x } ) )
22 uneq1 3572 . . . . . . . . . 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 3136 . . . . . . . 8  |-  ( ( ( d  \  {
x } )  e. 
~P b  /\  d  =  ( ( d 
\  { x }
)  u.  { x } ) )  ->  E. c  e.  ~P  b d  =  ( c  u.  { x } ) )
2515, 21, 24syl2anc 673 . . . . . . 7  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
265, 4elrnmpti 5091 . . . . . . 7  |-  ( d  e.  ran  F  <->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
2725, 26sylibr 217 . . . . . 6  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ran  F )
2827ssriv 3422 . . . . 5  |-  ( ~P ( b  u.  {
x } )  \  ~P b )  C_  ran  F
29 ssdomg 7633 . . . . 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 21 . . . 4  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  ~<_  ran  F )
31 domfi 7811 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ~P ( b  u.  { x }
)  \  ~P b
)  ~<_  ran  F )  ->  ( ~P ( b  u.  { x }
)  \  ~P b
)  e.  Fin )
3212, 30, 31syl2anc 673 . . 3  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  e.  Fin )
33 unfi 7856 . . 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 682 . 2  |-  ( ~P b  e.  Fin  ->  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)  e.  Fin )
351, 34syl5eqel 2553 1  |-  ( ~P b  e.  Fin  ->  ~P ( b  u.  {
x } )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   E.wrex 2757    \ cdif 3387    u. cun 3388    C_ wss 3390   ~Pcpw 3942   {csn 3959   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    Fn wfn 5584   -onto->wfo 5587    ~<_ cdom 7585   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-fin 7591
This theorem is referenced by:  pwfi  7887
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