Theorem rnco2 6212

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 6210	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5638	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2766	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

Syntax hints:   = wceq 1547  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  dmco  6213  isf34lem7  10299  isf34lem6  10300  imasless  17502  gsumzf1o  19885  gsumzmhm  19910  gsumzinv  19918  dprdf1o  20007  pf1rcl  22342  ovolficcss  25461  volsup  25548  uniiccdif  25570  uniioombllem3  25577  dyadmbl  25592  itg1climres  25706  cvmlift3lem6  35559  mblfinlem2  38032  volsupnfl  38039