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Theorem funimacnv 5324
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 5323 . . . 4  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
21rneqd 4906 . . 3  |-  ( Fun 
F  ->  ran  `' ( `' F  |`  A )  =  ran  ( F  |`  ( `' F " A ) ) )
3 df-ima 4702 . . 3  |-  ( F
" ( `' F " A ) )  =  ran  ( F  |`  ( `' F " A ) )
42, 3syl6reqr 2334 . 2  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ran  `' ( `' F  |`  A )
)
5 df-rn 4700 . . . 4  |-  ran  F  =  dom  `' F
65ineq2i 3367 . . 3  |-  ( A  i^i  ran  F )  =  ( A  i^i  dom  `' F )
7 dmres 4976 . . 3  |-  dom  ( `' F  |`  A )  =  ( A  i^i  dom  `' F )
8 dfdm4 4872 . . 3  |-  dom  ( `' F  |`  A )  =  ran  `' ( `' F  |`  A )
96, 7, 83eqtr2ri 2310 . 2  |-  ran  `' ( `' F  |`  A )  =  ( A  i^i  ran 
F )
104, 9syl6eq 2331 1  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    i^i cin 3151   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249
This theorem is referenced by:  funimass1  5325  funimass2  5326  isercolllem2  12139  isercolllem3  12140  isercoll  12141  cncls  17003  cvmliftlem15  23829  flfnei2  25577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
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