Theorem rncoss 4620

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 4619	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4374	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4538	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4554	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2101	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4374	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3038	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 2973   `'ccnv 4362   dom cdm 4363   ran crn 4364   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374

This theorem is referenced by:  cossxp  4863  fco  5076