Theorem dfrn4 5513

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 5051	. 2 ⊢ (𝐴 “ V) = ran (𝐴 ↾ V)
2		rnresv 5512	. 2 ⊢ ran (𝐴 ↾ V) = ran 𝐴
3	1, 2	eqtr2i 2633	1 ⊢ ran 𝐴 = (𝐴 “ V)

Syntax hints:   = wceq 1475  Vcvv 3173  ran crn 5039   ↾ cres 5040   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  csbrn  5514  dmmpt  5547  gsumpropd2lem  17096  ffsrn  28892