Theorem rncoss 4899

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 4898	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4639	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4814	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4830	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2198	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4639	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3198	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3131   `'ccnv 4627   dom cdm 4628   ran crn 4629   o. ccom 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639

This theorem is referenced by:  cossxp  5153  fco  5383  caseinj  7090  djuinj  7107