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

Theorem f1imacnv 5505
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 5003 . 2  |-  ( ( `' F  |`  ( F
" C ) )
" ( F " C ) )  =  ( `' F "
( F " C
) )
2 df-f1 5276 . . . . . . 7  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 450 . . . . . 6  |-  ( F : A -1-1-> B  ->  Fun  `' F )
43adantr 451 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' F
)
5 funcnvres 5337 . . . . 5  |-  ( Fun  `' F  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F " C
) ) )
64, 5syl 15 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F
" C ) ) )
76imaeq1d 5027 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  ( ( `' F  |`  ( F " C
) ) " ( F " C ) ) )
8 f1ores 5503 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
9 f1ocnv 5501 . . . . 5  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  ->  `' ( F  |`  C ) : ( F " C
)
-1-1-onto-> C )
108, 9syl 15 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C )
11 imadmrn 5040 . . . . 5  |-  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ran  `' ( F  |`  C )
12 f1odm 5492 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  dom  `' ( F  |`  C )  =  ( F " C ) )
1312imaeq2d 5028 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ( `' ( F  |`  C ) " ( F " C ) ) )
14 f1ofo 5495 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  `' ( F  |`  C ) : ( F " C
) -onto-> C )
15 forn 5470 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1614, 15syl 15 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1711, 13, 163eqtr3a 2352 . . . 4  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" ( F " C ) )  =  C )
1810, 17syl 15 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  C )
197, 18eqtr3d 2330 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( ( `' F  |`  ( F " C ) ) "
( F " C
) )  =  C )
201, 19syl5eqr 2342 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F " C
) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1opw2  6087  ssenen  7051  f1opwfi  7175  isf34lem3  8017  subggim  14746  gicsubgen  14758  cnt1  17094  basqtop  17418  tgqtop  17419  hmeoopn  17473  hmeocld  17474  hmeontr  17476  qtopf1  17523  ballotlemscr  23093  ballotlemrinv0  23107  tpr2rico  23311  cvmlift2lem9a  23849  f2imacnv  25155  oooeqim2  25156  grpokerinj  26678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
  Copyright terms: Public domain W3C validator