Theorem rnco2 6257

Description: The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Assertion

Ref	Expression
rnco2	⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

Proof of Theorem rnco2

Step	Hyp	Ref	Expression
1		rnco 6255	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5676	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2791	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  dmco  6258  isf34lem7  10374  isf34lem6  10375  imasless  17612  gsumzf1o  20006  gsumzmhm  20031  gsumzinv  20039  dprdf1o  20128  pf1rcl  22539  ovolficcss  25659  volsup  25746  uniiccdif  25768  uniioombllem3  25775  dyadmbl  25790  itg1climres  25904  cvmlift3lem6  35829  mblfinlem2  38342  volsupnfl  38349