Theorem rncoss 4874

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 4873	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4615	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4789	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4805	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2186	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4615	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3183	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3116   `'ccnv 4603   dom cdm 4604   ran crn 4605   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615

This theorem is referenced by:  cossxp  5126  fco  5353  caseinj  7054  djuinj  7071