Theorem rnresv 6205

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 6197	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	rneqi 5939	. 2 ⊢ ran ◡◡𝐴 = ran (𝐴 ↾ V)
3		rncnvcnv 5936	. 2 ⊢ ran ◡◡𝐴 = ran 𝐴
4	2, 3	eqtr3i 2758	1 ⊢ ran (𝐴 ↾ V) = ran 𝐴

Syntax hints:   = wceq 1534  Vcvv 3471  ^◡ccnv 5677  ran crn 5679   ↾ cres 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690

This theorem is referenced by:  dfrn4  6206  rnttrcl  9746