Theorem rncoss 5055

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 5054	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4787	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4967	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4984	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2259	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4787	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3289	1 |- ran ( A o. B ) C_ ran A

This proof depends on syntax axioms:   C_ wss 3220   `'ccnv 4775   dom cdm 4776   ran crn 4777   o. ccom 4780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787

This theorem is used by:  cossxp  5313  fco  5555  fcof  5897  caseinj  7435  djuinj  7452  znleval  15097