Theorem rnco2 6086

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 6085	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 5538	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2764	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

Syntax hints:   = wceq 1542  ran crn 5526   ↾ cres 5527   “ cima 5528   ∘ ccom 5529

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538

This theorem is referenced by:  dmco  6087  isf34lem7  9879  isf34lem6  9880  imasless  16916  gsumzf1o  19151  gsumzmhm  19176  gsumzinv  19184  dprdf1o  19273  pf1rcl  21119  ovolficcss  24221  volsup  24308  uniiccdif  24330  uniioombllem3  24337  dyadmbl  24352  itg1climres  24467  cvmlift3lem6  32857  mblfinlem2  35438  volsupnfl  35445