Theorem rnresv 5006

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 5000	. . 3 |- `' `' A = ( A |` _V )
2	1	rneqi 4775	. 2 |- ran `' `' A = ran ( A |` _V )
3		rncnvcnv 4772	. 2 |- ran `' `' A = ran A
4	2, 3	eqtr3i 2163	1 |- ran ( A |` _V ) = ran A

Syntax hints:   = wceq 1332   _Vcvv 2689   `'ccnv 4546   ran crn 4548   |` cres 4549

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559

This theorem is referenced by:  dfrn4  5007  casefun  6978