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Theorem funimacnv 5340
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 5339 . . . 4  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
21rneqd 4922 . . 3  |-  ( Fun 
F  ->  ran  `' ( `' F  |`  A )  =  ran  ( F  |`  ( `' F " A ) ) )
3 df-ima 4718 . . 3  |-  ( F
" ( `' F " A ) )  =  ran  ( F  |`  ( `' F " A ) )
42, 3syl6reqr 2347 . 2  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ran  `' ( `' F  |`  A )
)
5 df-rn 4716 . . . 4  |-  ran  F  =  dom  `' F
65ineq2i 3380 . . 3  |-  ( A  i^i  ran  F )  =  ( A  i^i  dom  `' F )
7 dmres 4992 . . 3  |-  dom  ( `' F  |`  A )  =  ( A  i^i  dom  `' F )
8 dfdm4 4888 . . 3  |-  dom  ( `' F  |`  A )  =  ran  `' ( `' F  |`  A )
96, 7, 83eqtr2ri 2323 . 2  |-  ran  `' ( `' F  |`  A )  =  ( A  i^i  ran 
F )
104, 9syl6eq 2344 1  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    i^i cin 3164   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265
This theorem is referenced by:  funimass1  5341  funimass2  5342  isercolllem2  12155  isercolllem3  12156  isercoll  12157  cncls  17019  cvmliftlem15  23844  flfnei2  25680
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
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