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Theorem isf34lem5 8258
Description: Lemma for isfin3-4 8262. (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 5216 . . . . . . 7  |-  ( F
" X )  C_  ran  F
2 compss.a . . . . . . . . . 10  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
32isf34lem2 8253 . . . . . . . . 9  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
43adantr 452 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
--> ~P A )
5 frn 5597 . . . . . . . 8  |-  ( F : ~P A --> ~P A  ->  ran  F  C_  ~P A )
64, 5syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ran  F  C_  ~P A )
71, 6syl5ss 3359 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  C_  ~P A )
8 simprl 733 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  ~P A )
9 fdm 5595 . . . . . . . . . . 11  |-  ( F : ~P A --> ~P A  ->  dom  F  =  ~P A )
104, 9syl 16 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  dom  F  =  ~P A )
118, 10sseqtr4d 3385 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  dom  F )
12 dfss1 3545 . . . . . . . . 9  |-  ( X 
C_  dom  F  <->  ( dom  F  i^i  X )  =  X )
1311, 12sylib 189 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =  X )
14 simprr 734 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
1513, 14eqnetrd 2619 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =/=  (/) )
16 imadisj 5223 . . . . . . . 8  |-  ( ( F " X )  =  (/)  <->  ( dom  F  i^i  X )  =  (/) )
1716necon3bii 2633 . . . . . . 7  |-  ( ( F " X )  =/=  (/)  <->  ( dom  F  i^i  X )  =/=  (/) )
1815, 17sylibr 204 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  =/=  (/) )
197, 18jca 519 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( ( F
" X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )
202isf34lem4 8257 . . . . 5  |-  ( ( A  e.  V  /\  ( ( F " X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
2119, 20syldan 457 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
222isf34lem3 8255 . . . . . 6  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
2322adantrr 698 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F "
( F " X
) )  =  X )
2423inteqd 4055 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  |^| ( F "
( F " X
) )  =  |^| X )
2521, 24eqtrd 2468 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| X )
2625fveq2d 5732 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  ( F `
 |^| X ) )
272compsscnv 8251 . . . 4  |-  `' F  =  F
2827fveq1i 5729 . . 3  |-  ( `' F `  ( F `
 U. ( F
" X ) ) )  =  ( F `
 ( F `  U. ( F " X
) ) )
292compssiso 8254 . . . . . 6  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
30 isof1o 6045 . . . . . 6  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  ->  F : ~P A -1-1-onto-> ~P A )
3129, 30syl 16 . . . . 5  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
3231adantr 452 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
-1-1-onto-> ~P A )
33 sspwuni 4176 . . . . . 6  |-  ( ( F " X ) 
C_  ~P A  <->  U. ( F " X )  C_  A )
347, 33sylib 189 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  C_  A
)
35 elpw2g 4363 . . . . . 6  |-  ( A  e.  V  ->  ( U. ( F " X
)  e.  ~P A  <->  U. ( F " X
)  C_  A )
)
3635adantr 452 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( U. ( F " X )  e. 
~P A  <->  U. ( F " X )  C_  A ) )
3734, 36mpbird 224 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  e.  ~P A )
38 f1ocnvfv1 6014 . . . 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 643 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( `' F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4028, 39syl5eqr 2482 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4126, 40eqtr3d 2470 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   U.cuni 4015   |^|cint 4050    e. cmpt 4266   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455   [ C.] crpss 6521
This theorem is referenced by:  isf34lem7  8259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-rpss 6522
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