Theorem rnco2 6274

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 6272	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5671	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2789	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ran crn 5659   ↾ cres 5660   “ cima 5661   ∘ ccom 5662

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by:  dmco  6275  isf34lem7  10496  isf34lem6  10497  imasless  17762  gsumzf1o  20177  gsumzmhm  20202  gsumzinv  20210  dprdf1o  20299  pf1rcl  22722  ovolficcss  25845  volsup  25932  uniiccdif  25954  uniioombllem3  25961  dyadmbl  25976  itg1climres  26090  acwer1prc  35893  cvmlift3lem6  36297  mblfinlem2  38844  volsupnfl  38851