MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1imacnv Structured version   Unicode version

Theorem f1imacnv 5683
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 5170 . 2  |-  ( ( `' F  |`  ( F
" C ) )
" ( F " C ) )  =  ( `' F "
( F " C
) )
2 df-f1 5451 . . . . . . 7  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 451 . . . . . 6  |-  ( F : A -1-1-> B  ->  Fun  `' F )
43adantr 452 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' F
)
5 funcnvres 5514 . . . . 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 5194 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  ( ( `' F  |`  ( F " C
) ) " ( F " C ) ) )
8 f1ores 5681 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
9 f1ocnv 5679 . . . . 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 5207 . . . . 5  |-  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ran  `' ( F  |`  C )
12 f1odm 5670 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  dom  `' ( F  |`  C )  =  ( F " C ) )
1312imaeq2d 5195 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ( `' ( F  |`  C ) " ( F " C ) ) )
14 f1ofo 5673 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  `' ( F  |`  C ) : ( F " C
) -onto-> C )
15 forn 5648 . . . . . 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 2491 . . . 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 2469 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( ( `' F  |`  ( F " C ) ) "
( F " C
) )  =  C )
201, 19syl5eqr 2481 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F " C
) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    C_ wss 3312   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445
This theorem is referenced by:  f1opw2  6290  ssenen  7273  f1opwfi  7402  isf34lem3  8247  subggim  15045  gicsubgen  15057  cnt1  17406  basqtop  17735  tgqtop  17736  hmeoopn  17790  hmeocld  17791  hmeontr  17793  qtopf1  17840  tpr2rico  24302  ballotlemscr  24768  ballotlemrinv0  24782  cvmlift2lem9a  24982  grpokerinj  26551
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
  Copyright terms: Public domain W3C validator