Theorem rncoss 4881

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

Assertion

Ref	Expression
rncoss	|- ran ( A o. B ) C_ ran A

Proof of Theorem rncoss

Step	Hyp	Ref	Expression
1		dmcoss 4880	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4622	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4796	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4812	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2191	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4622	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3188	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3121   `'ccnv 4610   dom cdm 4611   ran crn 4612   o. ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622

This theorem is referenced by:  cossxp  5133  fco  5363  caseinj  7066  djuinj  7083