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

Theorem isf34lem5 8797
Description: Lemma for isfin3-4 8801. (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 5190 . . . . . . 7  |-  ( F
" X )  C_  ran  F
2 compss.a . . . . . . . . . 10  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
32isf34lem2 8792 . . . . . . . . 9  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
43adantr 466 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
--> ~P A )
5 frn 5743 . . . . . . . 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 3472 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  C_  ~P A )
8 simprl 762 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  ~P A )
9 fdm 5741 . . . . . . . . . . 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 3498 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  C_  dom  F )
12 dfss1 3664 . . . . . . . . 9  |-  ( X 
C_  dom  F  <->  ( dom  F  i^i  X )  =  X )
1311, 12sylib 199 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =  X )
14 simprr 764 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
1513, 14eqnetrd 2715 . . . . . . 7  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( dom  F  i^i  X )  =/=  (/) )
16 imadisj 5198 . . . . . . . 8  |-  ( ( F " X )  =  (/)  <->  ( dom  F  i^i  X )  =  (/) )
1716necon3bii 2690 . . . . . . 7  |-  ( ( F " X )  =/=  (/)  <->  ( dom  F  i^i  X )  =/=  (/) )
1815, 17sylibr 215 . . . . . 6  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F " X )  =/=  (/) )
197, 18jca 534 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( ( F
" X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )
202isf34lem4 8796 . . . . 5  |-  ( ( A  e.  V  /\  ( ( F " X )  C_  ~P A  /\  ( F " X )  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
2119, 20syldan 472 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| ( F " ( F
" X ) ) )
222isf34lem3 8794 . . . . . 6  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
2322adantrr 721 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F "
( F " X
) )  =  X )
2423inteqd 4254 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  |^| ( F "
( F " X
) )  =  |^| X )
2521, 24eqtrd 2461 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. ( F " X
) )  =  |^| X )
2625fveq2d 5876 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  ( F `
 |^| X ) )
272compsscnv 8790 . . . 4  |-  `' F  =  F
2827fveq1i 5873 . . 3  |-  ( `' F `  ( F `
 U. ( F
" X ) ) )  =  ( F `
 ( F `  U. ( F " X
) ) )
292compssiso 8793 . . . . . 6  |-  ( A  e.  V  ->  F  Isom [
C.]  ,  `' [ C.]  ( ~P A ,  ~P A
) )
30 isof1o 6222 . . . . . 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 466 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  F : ~P A
-1-1-onto-> ~P A )
33 sspwuni 4382 . . . . . 6  |-  ( ( F " X ) 
C_  ~P A  <->  U. ( F " X )  C_  A )
347, 33sylib 199 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  C_  A
)
35 elpw2g 4579 . . . . . 6  |-  ( A  e.  V  ->  ( U. ( F " X
)  e.  ~P A  <->  U. ( F " X
)  C_  A )
)
3635adantr 466 . . . . 5  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( U. ( F " X )  e. 
~P A  <->  U. ( F " X )  C_  A ) )
3734, 36mpbird 235 . . . 4  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  U. ( F " X )  e.  ~P A )
38 f1ocnvfv1 6181 . . . 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 665 . . 3  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( `' F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4028, 39syl5eqr 2475 . 2  |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  ( F `  U. ( F " X ) ) )  =  U. ( F " X ) )
4126, 40eqtr3d 2463 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 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    i^i cin 3432    C_ wss 3433   (/)c0 3758   ~Pcpw 3976   U.cuni 4213   |^|cint 4249    |-> cmpt 4475   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848   -->wf 5588   -1-1-onto->wf1o 5591   ` cfv 5592    Isom wiso 5593   [ C.] crpss 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-rpss 6576
This theorem is referenced by:  isf34lem7  8798
  Copyright terms: Public domain W3C validator