Theorem rnresv 6143

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 6135	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	rneqi 5872	. 2 ⊢ ran ◡◡𝐴 = ran (𝐴 ↾ V)
3		rncnvcnv 5869	. 2 ⊢ ran ◡◡𝐴 = ran 𝐴
4	2, 3	eqtr3i 2756	1 ⊢ ran (𝐴 ↾ V) = ran 𝐴

Syntax hints:   = wceq 1541  Vcvv 3436  ^◡ccnv 5610  ran crn 5612   ↾ cres 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623

This theorem is referenced by:  dfrn4  6144  rnttrcl  9607