Theorem dfrn4 6026

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 5532	. 2 ⊢ (𝐴 “ V) = ran (𝐴 ↾ V)
2		rnresv 6025	. 2 ⊢ ran (𝐴 ↾ V) = ran 𝐴
3	1, 2	eqtr2i 2822	1 ⊢ ran 𝐴 = (𝐴 “ V)

Syntax hints:   = wceq 1538  Vcvv 3441  ran crn 5520   ↾ cres 5521   “ cima 5522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532

This theorem is referenced by:  csbrn  6027  dmmpt  6061  gsumpropd2lem  17881  ffsrn  30491