Theorem coexg 5174

Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Assertion

Ref	Expression
coexg	|- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )

Proof of Theorem coexg

Step	Hyp	Ref	Expression
1		cossxp 5152	. 2 |- ( A o. B ) C_ ( dom B X. ran A )
2		dmexg 4892	. . 3 |- ( B e. W -> dom B e. _V )
3		rnexg 4893	. . 3 |- ( A e. V -> ran A e. _V )
4		xpexg 4741	. . 3 |- ( ( dom B e. _V /\ ran A e. _V ) -> ( dom B X. ran A ) e. _V )
5	2, 3, 4	syl2anr 290	. 2 |- ( ( A e. V /\ B e. W ) -> ( dom B X. ran A ) e. _V )
6		ssexg 4143	. 2 |- ( ( ( A o. B ) C_ ( dom B X. ran A ) /\ ( dom B X. ran A ) e. _V ) -> ( A o. B ) e. _V )
7	1, 5, 6	sylancr 414	1 |- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   _Vcvv 2738   C_ wss 3130   X. cxp 4625   dom cdm 4627   ran crn 4628   o. ccom 4631

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

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-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638

This theorem is referenced by:  coex  5175  climcncf  14074