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Theorem f1imacnv 5832
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.)
Assertion
Ref Expression
f1imacnv  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F " C
) )  =  C )

Proof of Theorem f1imacnv
StepHypRef Expression
1 resima 5306 . 2  |-  ( ( `' F  |`  ( F
" C ) )
" ( F " C ) )  =  ( `' F "
( F " C
) )
2 df-f1 5593 . . . . . . 7  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 464 . . . . . 6  |-  ( F : A -1-1-> B  ->  Fun  `' F )
43adantr 465 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' F
)
5 funcnvres 5657 . . . . 5  |-  ( Fun  `' F  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F " C
) ) )
64, 5syl 16 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F
" C ) ) )
76imaeq1d 5336 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  ( ( `' F  |`  ( F " C
) ) " ( F " C ) ) )
8 f1ores 5830 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
9 f1ocnv 5828 . . . . 5  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  ->  `' ( F  |`  C ) : ( F " C
)
-1-1-onto-> C )
108, 9syl 16 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C )
11 imadmrn 5347 . . . . 5  |-  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ran  `' ( F  |`  C )
12 f1odm 5820 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  dom  `' ( F  |`  C )  =  ( F " C ) )
1312imaeq2d 5337 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ( `' ( F  |`  C ) " ( F " C ) ) )
14 f1ofo 5823 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  `' ( F  |`  C ) : ( F " C
) -onto-> C )
15 forn 5798 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1614, 15syl 16 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1711, 13, 163eqtr3a 2532 . . . 4  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" ( F " C ) )  =  C )
1810, 17syl 16 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  C )
197, 18eqtr3d 2510 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( ( `' F  |`  ( F " C ) ) "
( F " C
) )  =  C )
201, 19syl5eqr 2522 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F " C
) )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    C_ wss 3476   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582   -->wf 5584   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595
This theorem is referenced by:  f1opw2  6512  ssenen  7691  f1opwfi  7824  isf34lem3  8755  subggim  16119  gicsubgen  16131  cnt1  19645  basqtop  19975  tgqtop  19976  hmeoopn  20030  hmeocld  20031  hmeontr  20033  qtopf1  20080  f1otrg  23878  tpr2rico  27558  eulerpartlemmf  27982  ballotlemscr  28125  ballotlemrinv0  28139  cvmlift2lem9a  28416  grpokerinj  29978
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