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Theorem isf34lem5 8805
Description: Lemma for isfin3-4 8809. (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 5178 . . . . . . 7  |-  ( F
" X )  C_  ran  F
2 compss.a . . . . . . . . . 10  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
32isf34lem2 8800 . . . . . . . . 9  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
43adantr 467 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
--> ~P A )
5 frn 5733 . . . . . . . 8  |-  ( F : ~P A --> ~P A  ->  ran  F  C_  ~P A )
64, 5syl 17 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ran  F  C_  ~P A )
71, 6syl5ss 3442 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  C_  ~P A )
8 simprl 763 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  ~P A )
9 fdm 5731 . . . . . . . . . . 11  |-  ( F : ~P A --> ~P A  ->  dom  F  =  ~P A )
104, 9syl 17 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  dom  F  =  ~P A )
118, 10sseqtr4d 3468 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  dom  F )
12 dfss1 3636 . . . . . . . . 9  |-  ( X 
C_  dom  F  <->  ( dom  F  i^i  X )  =  X )
1311, 12sylib 200 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =  X )
14 simprr 765 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
1513, 14eqnetrd 2690 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =/=  (/) )
16 imadisj 5186 . . . . . . . 8  |-  ( ( F " X )  =  (/)  <->  ( dom  F  i^i  X )  =  (/) )
1716necon3bii 2675 . . . . . . 7  |-  ( ( F " X )  =/=  (/)  <->  ( dom  F  i^i  X )  =/=  (/) )
1815, 17sylibr 216 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  =/=  (/) )
197, 18jca 535 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( ( F
" X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )
202isf34lem4 8804 . . . . 5  |-  ( ( A  e.  V  /\  ( ( F " X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
2119, 20syldan 473 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
222isf34lem3 8802 . . . . . 6  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
2322adantrr 722 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F "
( F " X
) )  =  X )
2423inteqd 4238 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  |^| ( F "
( F " X
) )  =  |^| X )
2521, 24eqtrd 2484 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| X )
2625fveq2d 5867 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  ( F `
 |^| X ) )
272compsscnv 8798 . . . 4  |-  `' F  =  F
2827fveq1i 5864 . . 3  |-  ( `' F `  ( F `
 U. ( F
" X ) ) )  =  ( F `
 ( F `  U. ( F " X
) ) )
292compssiso 8801 . . . . . 6  |-  ( A  e.  V  ->  F  Isom [
C.]  ,  `' [ C.]  ( ~P A ,  ~P A
) )
30 isof1o 6214 . . . . . 6  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A
)  ->  F : ~P A -1-1-onto-> ~P A )
3129, 30syl 17 . . . . 5  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
3231adantr 467 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
-1-1-onto-> ~P A )
33 sspwuni 4366 . . . . . 6  |-  ( ( F " X ) 
C_  ~P A  <->  U. ( F " X )  C_  A )
347, 33sylib 200 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  C_  A
)
35 elpw2g 4565 . . . . . 6  |-  ( A  e.  V  ->  ( U. ( F " X
)  e.  ~P A  <->  U. ( F " X
)  C_  A )
)
3635adantr 467 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( U. ( F " X )  e. 
~P A  <->  U. ( F " X )  C_  A ) )
3734, 36mpbird 236 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  e.  ~P A )
38 f1ocnvfv1 6173 . . . 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 666 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( `' F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4028, 39syl5eqr 2498 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4126, 40eqtr3d 2486 1  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  |^| X )  =  U. ( F " X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621    \ cdif 3400    i^i cin 3402    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |^|cint 4233    |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581    Isom wiso 5582   [ C.] crpss 6567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-rpss 6568
This theorem is referenced by:  isf34lem7  8806
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