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Theorem pwfilem 7605
Description: Lemma for pwfi 7606. (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 4628 . 2  |-  ~P (
b  u.  { x } )  =  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)
2 vex 2975 . . . . . . . . 9  |-  c  e. 
_V
3 snex 4533 . . . . . . . . 9  |-  { x }  e.  _V
42, 3unex 6378 . . . . . . . 8  |-  ( c  u.  { x }
)  e.  _V
5 pwfilem.1 . . . . . . . 8  |-  F  =  ( c  e.  ~P b  |->  ( c  u. 
{ x } ) )
64, 5fnmpti 5539 . . . . . . 7  |-  F  Fn  ~P b
7 dffn4 5626 . . . . . . 7  |-  ( F  Fn  ~P b  <->  F : ~P b -onto-> ran  F )
86, 7mpbi 208 . . . . . 6  |-  F : ~P b -onto-> ran  F
9 fodomfi 7590 . . . . . 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 7534 . . . . 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 3478 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ~P ( b  u. 
{ x } ) )
143elpwun 6389 . . . . . . . . 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 3754 . . . . . . . . 9  |-  ( ( d  \  { x } )  u.  {
x } )  =  ( d  u.  {
x } )
17 elpwunsn 3917 . . . . . . . . . . 11  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  x  e.  d )
1817snssd 4018 . . . . . . . . . 10  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  { x }  C_  d )
19 ssequn2 3529 . . . . . . . . . 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 2488 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  =  ( ( d 
\  { x }
)  u.  { x } ) )
22 uneq1 3503 . . . . . . . . . 10  |-  ( c  =  ( d  \  { x } )  ->  ( c  u. 
{ x } )  =  ( ( d 
\  { x }
)  u.  { x } ) )
2322eqeq2d 2454 . . . . . . . . 9  |-  ( c  =  ( d  \  { x } )  ->  ( d  =  ( c  u.  {
x } )  <->  d  =  ( ( d  \  { x } )  u.  { x }
) ) )
2423rspcev 3073 . . . . . . . 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 5090 . . . . . . 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 3360 . . . . 5  |-  ( ~P ( b  u.  {
x } )  \  ~P b )  C_  ran  F
29 ssdomg 7355 . . . . 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 7534 . . . 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 7579 . . 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 2527 1  |-  ( ~P b  e.  Fin  ->  ~P ( b  u.  {
x } )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   E.wrex 2716    \ cdif 3325    u. cun 3326    C_ wss 3328   ~Pcpw 3860   {csn 3877   class class class wbr 4292    e. cmpt 4350   ran crn 4841    Fn wfn 5413   -onto->wfo 5416    ~<_ cdom 7308   Fincfn 7310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-fin 7314
This theorem is referenced by:  pwfi  7606
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