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Theorem rnresv 5047
Description: The range of a universal restriction. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
rnresv  |-  ran  ( A  |`  _V )  =  ran  A

Proof of Theorem rnresv
StepHypRef Expression
1 cnvcnv2 5041 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21rneqi 4816 . 2  |-  ran  `' `' A  =  ran  ( A  |`  _V )
3 rncnvcnv 4813 . 2  |-  ran  `' `' A  =  ran  A
42, 3eqtr3i 2180 1  |-  ran  ( A  |`  _V )  =  ran  A
Colors of variables: wff set class
Syntax hints:    = wceq 1335   _Vcvv 2712   `'ccnv 4587   ran crn 4589    |` cres 4590
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600
This theorem is referenced by:  dfrn4  5048  casefun  7031
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