Theorem rncoss 4933

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 4932	. 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴
2		df-rn 4671	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
3		cnvco 4848	. . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
4	3	dmeqi 4864	. . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
5	2, 4	eqtri 2214	. 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
6		df-rn 4671	. 2 ⊢ ran 𝐴 = dom ◡𝐴
7	1, 5, 6	3sstr4i 3221	1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3154  ^◡ccnv 4659  dom cdm 4660  ran crn 4661   ∘ ccom 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671

This theorem is referenced by:  cossxp  5189  fco  5420  caseinj  7150  djuinj  7167  znleval  14152