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Theorem f1imacnv 5480
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 4942 . 2  |-  ( ( `' F  |`  ( F
" C ) )
" ( F " C ) )  =  ( `' F "
( F " C
) )
2 df-f1 5223 . . . . . . 7  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 275 . . . . . 6  |-  ( F : A -1-1-> B  ->  Fun  `' F )
43adantr 276 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  Fun  `' F
)
5 funcnvres 5291 . . . . 5  |-  ( Fun  `' F  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F " C
) ) )
64, 5syl 14 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C )  =  ( `' F  |`  ( F
" C ) ) )
76imaeq1d 4971 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  ( ( `' F  |`  ( F " C
) ) " ( F " C ) ) )
8 f1ores 5478 . . . . 5  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
9 f1ocnv 5476 . . . . 5  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  ->  `' ( F  |`  C ) : ( F " C
)
-1-1-onto-> C )
108, 9syl 14 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C )
11 imadmrn 4982 . . . . 5  |-  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ran  `' ( F  |`  C )
12 f1odm 5467 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  dom  `' ( F  |`  C )  =  ( F " C ) )
1312imaeq2d 4972 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" dom  `' ( F  |`  C ) )  =  ( `' ( F  |`  C ) " ( F " C ) ) )
14 f1ofo 5470 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  `' ( F  |`  C ) : ( F " C
) -onto-> C )
15 forn 5443 . . . . . 6  |-  ( `' ( F  |`  C ) : ( F " C ) -onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1614, 15syl 14 . . . . 5  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ran  `' ( F  |`  C )  =  C )
1711, 13, 163eqtr3a 2234 . . . 4  |-  ( `' ( F  |`  C ) : ( F " C ) -1-1-onto-> C  ->  ( `' ( F  |`  C )
" ( F " C ) )  =  C )
1810, 17syl 14 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' ( F  |`  C ) " ( F " C ) )  =  C )
197, 18eqtr3d 2212 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( ( `' F  |`  ( F " C ) ) "
( F " C
) )  =  C )
201, 19eqtr3id 2224 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F " C
) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    C_ wss 3131   `'ccnv 4627   dom cdm 4628   ran crn 4629    |` cres 4630   "cima 4631   Fun wfun 5212   -->wf 5214   -1-1->wf1 5215   -onto->wfo 5216   -1-1-onto->wf1o 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225
This theorem is referenced by:  f1opw2  6079  ssenen  6853  hmeoopn  13896  hmeocld  13897  hmeontr  13898
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