Theorem rncoss 5926

Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)

Assertion

Ref	Expression
rncoss	⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴

Proof of Theorem rncoss

Step	Hyp	Ref	Expression
1		dmcoss 5924	. 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴
2		df-rn 5636	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
3		cnvco 5834	. . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
4	3	dmeqi 5853	. . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
5	2, 4	eqtri 2763	. 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
6		df-rn 5636	. 2 ⊢ ran 𝐴 = dom ◡𝐴
7	1, 5, 6	3sstr4i 3973	1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3890  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ∘ 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-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-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636

This theorem is referenced by:  cossxp  6230  fcof  6685  fin23lem29  10261  fin23lem30  10262  wunco  10654  imasless  17502  gsumzf1o  19885  znleval  21536  pi1xfrcnvlem  25048  pjss1coi  32259  pj3i  32304  smatrcl  33987  mblfinlem3  38033  mblfinlem4  38034  ismblfin  38035  relexp0a  44167  rntrclfv  44183  stoweidlem27  46477  fourierdlem42  46599  hoicvr  46998