Theorem rnresv 6200

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 6191	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	rneqi 5927	. 2 ⊢ ran ◡◡𝐴 = ran (𝐴 ↾ V)
3		rncnvcnv 5924	. 2 ⊢ ran ◡◡𝐴 = ran 𝐴
4	2, 3	eqtr3i 2788	1 ⊢ ran (𝐴 ↾ V) = ran 𝐴

Syntax hints:   = wceq 1570  Vcvv 3455  ^◡ccnv 5660  ran crn 5662   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673

This theorem is referenced by:  dfrn4  6201  imadifssran  6202  rnttrcl  9687