Theorem rncoss 4946

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 4945	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4684	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4861	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4877	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2225	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4684	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3233	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3165   `'ccnv 4672   dom cdm 4673   ran crn 4674   o. ccom 4677

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684

This theorem is referenced by:  cossxp  5202  fco  5435  caseinj  7173  djuinj  7190  znleval  14333