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Theorem fimacnvinrn 24000
Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
fimacnvinrn  |-  ( Fun 
F  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )

Proof of Theorem fimacnvinrn
StepHypRef Expression
1 inpreima 5816 . 2  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
2 funforn 5619 . . . . 5  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
3 fof 5612 . . . . 5  |-  ( F : dom  F -onto-> ran  F  ->  F : dom  F --> ran  F )
42, 3sylbi 188 . . . 4  |-  ( Fun 
F  ->  F : dom  F --> ran  F )
5 fimacnv 5821 . . . 4  |-  ( F : dom  F --> ran  F  ->  ( `' F " ran  F )  =  dom  F )
64, 5syl 16 . . 3  |-  ( Fun 
F  ->  ( `' F " ran  F )  =  dom  F )
76ineq2d 3502 . 2  |-  ( Fun 
F  ->  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( ( `' F " A )  i^i  dom  F )
)
8 cnvresima 5318 . . 3  |-  ( `' ( F  |`  dom  F
) " A )  =  ( ( `' F " A )  i^i  dom  F )
9 resdm2 5319 . . . . . 6  |-  ( F  |`  dom  F )  =  `' `' F
10 funrel 5430 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
11 dfrel2 5280 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
1210, 11sylib 189 . . . . . 6  |-  ( Fun 
F  ->  `' `' F  =  F )
139, 12syl5eq 2448 . . . . 5  |-  ( Fun 
F  ->  ( F  |` 
dom  F )  =  F )
1413cnveqd 5007 . . . 4  |-  ( Fun 
F  ->  `' ( F  |`  dom  F )  =  `' F )
1514imaeq1d 5161 . . 3  |-  ( Fun 
F  ->  ( `' ( F  |`  dom  F
) " A )  =  ( `' F " A ) )
168, 15syl5eqr 2450 . 2  |-  ( Fun 
F  ->  ( ( `' F " A )  i^i  dom  F )  =  ( `' F " A ) )
171, 7, 163eqtrrd 2441 1  |-  ( Fun 
F  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    i^i cin 3279   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Rel wrel 4842   Fun wfun 5407   -->wf 5409   -onto->wfo 5411
This theorem is referenced by:  fimacnvinrn2  24001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421
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