Theorem rncoss 4937

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 4936	. 2 |- dom ( `' B o. `' A ) C_ dom `' A
2		df-rn 4675	. . 3 |- ran ( A o. B ) = dom `' ( A o. B )
3		cnvco 4852	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
4	3	dmeqi 4868	. . 3 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
5	2, 4	eqtri 2217	. 2 |- ran ( A o. B ) = dom ( `' B o. `' A )
6		df-rn 4675	. 2 |- ran A = dom `' A
7	1, 5, 6	3sstr4i 3225	1 |- ran ( A o. B ) C_ ran A

Syntax hints:   C_ wss 3157   `'ccnv 4663   dom cdm 4664   ran crn 4665   o. ccom 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675

This theorem is referenced by:  cossxp  5193  fco  5426  caseinj  7164  djuinj  7181  znleval  14285