Theorem rncoss 4958

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 4957	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4694	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4871	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4888	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2227	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4694	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3238	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3170   `'ccnv 4682   dom cdm 4683   ran crn 4684   o. ccom 4687

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694

This theorem is referenced by:  cossxp  5214  fco  5451  caseinj  7206  djuinj  7223  znleval  14490