Theorem dfrn4 6202

Description: Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Assertion

Ref	Expression
dfrn4	⊢ ran 𝐴 = (𝐴 “ V)

Proof of Theorem dfrn4

Step	Hyp	Ref	Expression
1		df-ima 5675	. 2 ⊢ (𝐴 “ V) = ran (𝐴 ↾ V)
2		rnresv 6201	. 2 ⊢ ran (𝐴 ↾ V) = ran 𝐴
3	1, 2	eqtr2i 2793	1 ⊢ ran 𝐴 = (𝐴 “ V)

Syntax hints:   = wceq 1567  Vcvv 3463  ran crn 5663   ↾ cres 5664   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  csbrn  6205  dmmpt  6242  gsumpropd2lem  18736  ffsrn  33013