MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf34lem5 Unicode version

Theorem isf34lem5 8020
Description: Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
isf34lem5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  |^| X )  =  U. ( F " X ) )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    F( x)    X( x)

Proof of Theorem isf34lem5
StepHypRef Expression
1 imassrn 5041 . . . . . . 7  |-  ( F
" X )  C_  ran  F
2 compss.a . . . . . . . . . 10  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
32isf34lem2 8015 . . . . . . . . 9  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
43adantr 451 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
--> ~P A )
5 frn 5411 . . . . . . . 8  |-  ( F : ~P A --> ~P A  ->  ran  F  C_  ~P A )
64, 5syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ran  F  C_  ~P A )
71, 6syl5ss 3203 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  C_  ~P A )
8 simprl 732 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  ~P A )
9 fdm 5409 . . . . . . . . . . 11  |-  ( F : ~P A --> ~P A  ->  dom  F  =  ~P A )
104, 9syl 15 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  dom  F  =  ~P A )
118, 10sseqtr4d 3228 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  dom  F )
12 dfss1 3386 . . . . . . . . 9  |-  ( X 
C_  dom  F  <->  ( dom  F  i^i  X )  =  X )
1311, 12sylib 188 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =  X )
14 simprr 733 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
1513, 14eqnetrd 2477 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =/=  (/) )
16 imadisj 5048 . . . . . . . 8  |-  ( ( F " X )  =  (/)  <->  ( dom  F  i^i  X )  =  (/) )
1716necon3bii 2491 . . . . . . 7  |-  ( ( F " X )  =/=  (/)  <->  ( dom  F  i^i  X )  =/=  (/) )
1815, 17sylibr 203 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  =/=  (/) )
197, 18jca 518 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( ( F
" X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )
202isf34lem4 8019 . . . . 5  |-  ( ( A  e.  V  /\  ( ( F " X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
2119, 20syldan 456 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
222isf34lem3 8017 . . . . . 6  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
2322adantrr 697 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F "
( F " X
) )  =  X )
2423inteqd 3883 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  |^| ( F "
( F " X
) )  =  |^| X )
2521, 24eqtrd 2328 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| X )
2625fveq2d 5545 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  ( F `
 |^| X ) )
272compsscnv 8013 . . . 4  |-  `' F  =  F
2827fveq1i 5542 . . 3  |-  ( `' F `  ( F `
 U. ( F
" X ) ) )  =  ( F `
 ( F `  U. ( F " X
) ) )
292compssiso 8016 . . . . . 6  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
30 isof1o 5838 . . . . . 6  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  ->  F : ~P A -1-1-onto-> ~P A )
3129, 30syl 15 . . . . 5  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
3231adantr 451 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
-1-1-onto-> ~P A )
33 sspwuni 4003 . . . . . 6  |-  ( ( F " X ) 
C_  ~P A  <->  U. ( F " X )  C_  A )
347, 33sylib 188 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  C_  A
)
35 elpw2g 4190 . . . . . 6  |-  ( A  e.  V  ->  ( U. ( F " X
)  e.  ~P A  <->  U. ( F " X
)  C_  A )
)
3635adantr 451 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( U. ( F " X )  e. 
~P A  <->  U. ( F " X )  C_  A ) )
3734, 36mpbird 223 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  e.  ~P A )
38 f1ocnvfv1 5808 . . . 4  |-  ( ( F : ~P A -1-1-onto-> ~P A  /\  U. ( F
" X )  e. 
~P A )  -> 
( `' F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
3932, 37, 38syl2anc 642 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( `' F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4028, 39syl5eqr 2342 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4126, 40eqtr3d 2330 1  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  |^| X )  =  U. ( F " X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   [ C.] crpss 6292
This theorem is referenced by:  isf34lem7  8021
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  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-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-isom 5280  df-rpss 6293
  Copyright terms: Public domain W3C validator