Theorem rnco2 5195

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 5194	. 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
2		df-ima 4692	. 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
3	1, 2	eqtr4i 2230	1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)

Syntax hints:   = wceq 1373  ran crn 4680   ↾ cres 4681   “ cima 4682   ∘ ccom 4683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692

This theorem is referenced by:  dmco  5196