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 5637	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2763	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

Syntax hints:   = wceq 1542  ran crn 5625   ↾ cres 5626   “ cima 5627   ∘ ccom 5628

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  dmco  6213  isf34lem7  10292  isf34lem6  10293  imasless  17495  gsumzf1o  19878  gsumzmhm  19903  gsumzinv  19911  dprdf1o  20000  pf1rcl  22324  ovolficcss  25446  volsup  25533  uniiccdif  25555  uniioombllem3  25562  dyadmbl  25577  itg1climres  25691  cvmlift3lem6  35522  mblfinlem2  37993  volsupnfl  38000