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Theorem funimacnv 5599
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 5598 . . . 4  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
21rneqd 5176 . . 3  |-  ( Fun 
F  ->  ran  `' ( `' F  |`  A )  =  ran  ( F  |`  ( `' F " A ) ) )
3 df-ima 4962 . . 3  |-  ( F
" ( `' F " A ) )  =  ran  ( F  |`  ( `' F " A ) )
42, 3syl6reqr 2514 . 2  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ran  `' ( `' F  |`  A )
)
5 df-rn 4960 . . . 4  |-  ran  F  =  dom  `' F
65ineq2i 3658 . . 3  |-  ( A  i^i  ran  F )  =  ( A  i^i  dom  `' F )
7 dmres 5240 . . 3  |-  dom  ( `' F  |`  A )  =  ( A  i^i  dom  `' F )
8 dfdm4 5141 . . 3  |-  dom  ( `' F  |`  A )  =  ran  `' ( `' F  |`  A )
96, 7, 83eqtr2ri 2490 . 2  |-  ran  `' ( `' F  |`  A )  =  ( A  i^i  ran 
F )
104, 9syl6eq 2511 1  |-  ( Fun 
F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    i^i cin 3436   `'ccnv 4948   dom cdm 4949   ran crn 4950    |` cres 4951   "cima 4952   Fun wfun 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-fun 5529
This theorem is referenced by:  funimass1  5600  funimass2  5601  isercolllem2  13262  isercolllem3  13263  isercoll  13264  cncls  19011  ffsrn  26181  cvmliftlem15  27332
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