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Theorem funimacnv 5169
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 5168 . . . 4  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
21rneqd 4738 . . 3  |-  ( Fun 
F  ->  ran  `' ( `' F  |`  A )  =  ran  ( F  |`  ( `' F " A ) ) )
3 df-ima 4522 . . 3  |-  ( F
" ( `' F " A ) )  =  ran  ( F  |`  ( `' F " A ) )
42, 3syl6reqr 2169 . 2  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ran  `' ( `' F  |`  A )
)
5 df-rn 4520 . . . 4  |-  ran  F  =  dom  `' F
65ineq2i 3244 . . 3  |-  ( A  i^i  ran  F )  =  ( A  i^i  dom  `' F )
7 dmres 4810 . . 3  |-  dom  ( `' F  |`  A )  =  ( A  i^i  dom  `' F )
8 dfdm4 4701 . . 3  |-  dom  ( `' F  |`  A )  =  ran  `' ( `' F  |`  A )
96, 7, 83eqtr2ri 2145 . 2  |-  ran  `' ( `' F  |`  A )  =  ( A  i^i  ran 
F )
104, 9syl6eq 2166 1  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    i^i cin 3040   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511   "cima 4512   Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095
This theorem is referenced by:  funimass1  5170  funimass2  5171
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