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Theorem funimacnv 5517
Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimacnv  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )

Proof of Theorem funimacnv
StepHypRef Expression
1 funcnvres2 5516 . . . 4  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
21rneqd 5089 . . 3  |-  ( Fun 
F  ->  ran  `' ( `' F  |`  A )  =  ran  ( F  |`  ( `' F " A ) ) )
3 df-ima 4883 . . 3  |-  ( F
" ( `' F " A ) )  =  ran  ( F  |`  ( `' F " A ) )
42, 3syl6reqr 2486 . 2  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ran  `' ( `' F  |`  A )
)
5 df-rn 4881 . . . 4  |-  ran  F  =  dom  `' F
65ineq2i 3531 . . 3  |-  ( A  i^i  ran  F )  =  ( A  i^i  dom  `' F )
7 dmres 5159 . . 3  |-  dom  ( `' F  |`  A )  =  ( A  i^i  dom  `' F )
8 dfdm4 5055 . . 3  |-  dom  ( `' F  |`  A )  =  ran  `' ( `' F  |`  A )
96, 7, 83eqtr2ri 2462 . 2  |-  ran  `' ( `' F  |`  A )  =  ( A  i^i  ran 
F )
104, 9syl6eq 2483 1  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    i^i cin 3311   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440
This theorem is referenced by:  funimass1  5518  funimass2  5519  isercolllem2  12451  isercolllem3  12452  isercoll  12453  cncls  17330  cvmliftlem15  24977
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
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