Theorem dfrn4 6157

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 5634	. 2 ⊢ (𝐴 “ V) = ran (𝐴 ↾ V)
2		rnresv 6156	. 2 ⊢ ran (𝐴 ↾ V) = ran 𝐴
3	1, 2	eqtr2i 2757	1 ⊢ ran 𝐴 = (𝐴 “ V)

Syntax hints:   = wceq 1541  Vcvv 3437  ran crn 5622   ↾ cres 5623   “ cima 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by:  csbrn  6158  dmmpt  6195  gsumpropd2lem  18595  ffsrn  32735