Theorem rnco2 6254

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 6252	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5664	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2787	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655

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 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  dmco  6255  isf34lem7  10450  isf34lem6  10451  imasless  17705  gsumzf1o  20119  gsumzmhm  20144  gsumzinv  20152  dprdf1o  20241  pf1rcl  22660  ovolficcss  25783  volsup  25870  uniiccdif  25892  uniioombllem3  25899  dyadmbl  25914  itg1climres  26028  cvmlift3lem6  36068  mblfinlem2  38556  volsupnfl  38563