Theorem rnco2 6250

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 6248	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5668	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2786	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ran crn 5656   ↾ cres 5657   “ cima 5658   ∘ ccom 5659

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  dmco  6251  isf34lem7  10381  isf34lem6  10382  imasless  17626  gsumzf1o  20039  gsumzmhm  20064  gsumzinv  20072  dprdf1o  20161  pf1rcl  22574  ovolficcss  25697  volsup  25784  uniiccdif  25806  uniioombllem3  25813  dyadmbl  25828  itg1climres  25942  cvmlift3lem6  35903  mblfinlem2  38407  volsupnfl  38414