Theorem rnresv 5100

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

Step	Hyp	Ref	Expression
1		cnvcnv2 5094	. . 3 |- `' `' A = ( A |` _V )
2	1	rneqi 4867	. 2 |- ran `' `' A = ran ( A |` _V )
3		rncnvcnv 4864	. 2 |- ran `' `' A = ran A
4	2, 3	eqtr3i 2210	1 |- ran ( A |` _V ) = ran A

Syntax hints:   = wceq 1363   _Vcvv 2749   `'ccnv 4637   ran crn 4639   |` cres 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650

This theorem is referenced by:  dfrn4  5101  casefun  7098