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Theorem rnresv 5629
 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
StepHypRef Expression
1 cnvcnv2 5623 . . 3 𝐴 = (𝐴 ↾ V)
21rneqi 5384 . 2 ran 𝐴 = ran (𝐴 ↾ V)
3 rncnvcnv 5381 . 2 ran 𝐴 = ran 𝐴
42, 3eqtr3i 2675 1 ran (𝐴 ↾ V) = ran 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523  Vcvv 3231  ◡ccnv 5142  ran crn 5144   ↾ cres 5145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155 This theorem is referenced by:  dfrn4  5630
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