Theorem rnresv 6174

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 6166	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	rneqi 5901	. 2 ⊢ ran ◡◡𝐴 = ran (𝐴 ↾ V)
3		rncnvcnv 5898	. 2 ⊢ ran ◡◡𝐴 = ran 𝐴
4	2, 3	eqtr3i 2754	1 ⊢ ran (𝐴 ↾ V) = ran 𝐴

Syntax hints:   = wceq 1540  Vcvv 3447  ^◡ccnv 5637  ran crn 5639   ↾ cres 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650

This theorem is referenced by:  dfrn4  6175  rnttrcl  9675