Theorem rncoss 4936

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 4935	. 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴
2		df-rn 4674	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
3		cnvco 4851	. . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
4	3	dmeqi 4867	. . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
5	2, 4	eqtri 2217	. 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
6		df-rn 4674	. 2 ⊢ ran 𝐴 = dom ◡𝐴
7	1, 5, 6	3sstr4i 3224	1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3157  ^◡ccnv 4662  dom cdm 4663  ran crn 4664   ∘ ccom 4667

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674

This theorem is referenced by:  cossxp  5192  fco  5423  caseinj  7155  djuinj  7172  znleval  14209