Theorem dfrn4 6189

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 5660	. 2 ⊢ (𝐴 “ V) = ran (𝐴 ↾ V)
2		rnresv 6188	. 2 ⊢ ran (𝐴 ↾ V) = ran 𝐴
3	1, 2	eqtr2i 2786	1 ⊢ ran 𝐴 = (𝐴 “ V)

Syntax hints:   = wceq 1560  Vcvv 3454  ran crn 5648   ↾ cres 5649   “ cima 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  csbrn  6190  dmmpt  6227  gsumpropd2lem  18713  ffsrn  32927