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

Theorem isf34lem5 8004
Description: Lemma for isfin3-4 8008. (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 5025 . . . . . . 7  |-  ( F
" X )  C_  ran  F
2 compss.a . . . . . . . . . 10  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
32isf34lem2 7999 . . . . . . . . 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 5395 . . . . . . . 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 3190 . . . . . 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 5393 . . . . . . . . . . 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 3215 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  dom  F )
12 dfss1 3373 . . . . . . . . 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 2464 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =/=  (/) )
16 imadisj 5032 . . . . . . . 8  |-  ( ( F " X )  =  (/)  <->  ( dom  F  i^i  X )  =  (/) )
1716necon3bii 2478 . . . . . . 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 8003 . . . . 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 8001 . . . . . 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 3867 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  |^| ( F "
( F " X
) )  =  |^| X )
2521, 24eqtrd 2315 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| X )
2625fveq2d 5529 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  ( F `
 |^| X ) )
272compsscnv 7997 . . . 4  |-  `' F  =  F
2827fveq1i 5526 . . 3  |-  ( `' F `  ( F `
 U. ( F
" X ) ) )  =  ( F `
 ( F `  U. ( F " X
) ) )
292compssiso 8000 . . . . . 6  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
30 isof1o 5822 . . . . . 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 3987 . . . . . 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 4174 . . . . . 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 5792 . . . 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 2329 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4126, 40eqtr3d 2317 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   [ C.] crpss 6276
This theorem is referenced by:  isf34lem7  8005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-rpss 6277
  Copyright terms: Public domain W3C validator