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Theorem pwfilem 7166
Description: Lemma for pwfi 7167. (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 4316 . 2  |-  ~P (
b  u.  { x } )  =  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)
2 vex 2804 . . . . . . . . 9  |-  c  e. 
_V
3 snex 4232 . . . . . . . . 9  |-  { x }  e.  _V
42, 3unex 4534 . . . . . . . 8  |-  ( c  u.  { x }
)  e.  _V
5 pwfilem.1 . . . . . . . 8  |-  F  =  ( c  e.  ~P b  |->  ( c  u. 
{ x } ) )
64, 5fnmpti 5388 . . . . . . 7  |-  F  Fn  ~P b
7 dffn4 5473 . . . . . . 7  |-  ( F  Fn  ~P b  <->  F : ~P b -onto-> ran  F )
86, 7mpbi 199 . . . . . 6  |-  F : ~P b -onto-> ran  F
9 fodomfi 7151 . . . . . 6  |-  ( ( ~P b  e.  Fin  /\  F : ~P b -onto-> ran  F )  ->  ran  F  ~<_  ~P b )
108, 9mpan2 652 . . . . 5  |-  ( ~P b  e.  Fin  ->  ran 
F  ~<_  ~P b )
11 domfi 7100 . . . . 5  |-  ( ( ~P b  e.  Fin  /\ 
ran  F  ~<_  ~P b
)  ->  ran  F  e. 
Fin )
1210, 11mpdan 649 . . . 4  |-  ( ~P b  e.  Fin  ->  ran 
F  e.  Fin )
13 eldifi 3311 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ~P ( b  u. 
{ x } ) )
143elpwun 4583 . . . . . . . . 9  |-  ( d  e.  ~P ( b  u.  { x }
)  <->  ( d  \  { x } )  e.  ~P b )
1513, 14sylib 188 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  \  { x } )  e.  ~P b )
16 undif1 3542 . . . . . . . . 9  |-  ( ( d  \  { x } )  u.  {
x } )  =  ( d  u.  {
x } )
17 elpwunsn 4584 . . . . . . . . . . 11  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  x  e.  d )
1817snssd 3776 . . . . . . . . . 10  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  { x }  C_  d )
19 ssequn2 3361 . . . . . . . . . 10  |-  ( { x }  C_  d  <->  ( d  u.  { x } )  =  d )
2018, 19sylib 188 . . . . . . . . 9  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  (
d  u.  { x } )  =  d )
2116, 20syl5req 2341 . . . . . . . 8  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  =  ( ( d 
\  { x }
)  u.  { x } ) )
22 uneq1 3335 . . . . . . . . . 10  |-  ( c  =  ( d  \  { x } )  ->  ( c  u. 
{ x } )  =  ( ( d 
\  { x }
)  u.  { x } ) )
2322eqeq2d 2307 . . . . . . . . 9  |-  ( c  =  ( d  \  { x } )  ->  ( d  =  ( c  u.  {
x } )  <->  d  =  ( ( d  \  { x } )  u.  { x }
) ) )
2423rspcev 2897 . . . . . . . 8  |-  ( ( ( d  \  {
x } )  e. 
~P b  /\  d  =  ( ( d 
\  { x }
)  u.  { x } ) )  ->  E. c  e.  ~P  b d  =  ( c  u.  { x } ) )
2515, 21, 24syl2anc 642 . . . . . . 7  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
265, 4elrnmpti 4946 . . . . . . 7  |-  ( d  e.  ran  F  <->  E. c  e.  ~P  b d  =  ( c  u.  {
x } ) )
2725, 26sylibr 203 . . . . . 6  |-  ( d  e.  ( ~P (
b  u.  { x } )  \  ~P b )  ->  d  e.  ran  F )
2827ssriv 3197 . . . . 5  |-  ( ~P ( b  u.  {
x } )  \  ~P b )  C_  ran  F
29 ssdomg 6923 . . . . 5  |-  ( ran 
F  e.  Fin  ->  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  C_  ran  F  -> 
( ~P ( b  u.  { x }
)  \  ~P b
)  ~<_  ran  F )
)
3012, 28, 29ee10 1366 . . . 4  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  ~<_  ran  F )
31 domfi 7100 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ~P ( b  u.  { x }
)  \  ~P b
)  ~<_  ran  F )  ->  ( ~P ( b  u.  { x }
)  \  ~P b
)  e.  Fin )
3212, 30, 31syl2anc 642 . . 3  |-  ( ~P b  e.  Fin  ->  ( ~P ( b  u. 
{ x } ) 
\  ~P b )  e.  Fin )
33 unfi 7140 . . 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 650 . 2  |-  ( ~P b  e.  Fin  ->  ( ( ~P ( b  u.  { x }
)  \  ~P b
)  u.  ~P b
)  e.  Fin )
351, 34syl5eqel 2380 1  |-  ( ~P b  e.  Fin  ->  ~P ( b  u.  {
x } )  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162    u. cun 3163    C_ wss 3165   ~Pcpw 3638   {csn 3653   class class class wbr 4039    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -onto->wfo 5269    ~<_ cdom 6877   Fincfn 6879
This theorem is referenced by:  pwfi  7167
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-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-int 3879  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883
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