Theorem rncoss 4994

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 4993	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4729	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4906	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4923	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2250	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4729	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3265	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3197   `'ccnv 4717   dom cdm 4718   ran crn 4719   o. ccom 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729

This theorem is referenced by:  cossxp  5250  fco  5488  caseinj  7252  djuinj  7269  znleval  14611