Theorem rnresv 6188

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 6179	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	rneqi 5913	. 2 ⊢ ran ◡◡𝐴 = ran (𝐴 ↾ V)
3		rncnvcnv 5910	. 2 ⊢ ran ◡◡𝐴 = ran 𝐴
4	2, 3	eqtr3i 2787	1 ⊢ ran (𝐴 ↾ V) = ran 𝐴

Syntax hints:   = wceq 1560  Vcvv 3454  ^◡ccnv 5646  ran crn 5648   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659

This theorem is referenced by:  dfrn4  6189  rnttrcl  9677