Theorem cofunexg 6112

Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cofunexg	|- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )

Proof of Theorem cofunexg

Step	Hyp	Ref	Expression
1		relco 5129	. . 3 |- Rel ( A o. B )
2		relssdmrn 5151	. . 3 |- ( Rel ( A o. B ) -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
3	1, 2	ax-mp 5	. 2 |- ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) )
4		dmcoss 4898	. . . . 5 |- dom ( A o. B ) C_ dom B
5		dmexg 4893	. . . . 5 |- ( B e. C -> dom B e. _V )
6		ssexg 4144	. . . . 5 |- ( ( dom ( A o. B ) C_ dom B /\ dom B e. _V ) -> dom ( A o. B ) e. _V )
7	4, 5, 6	sylancr 414	. . . 4 |- ( B e. C -> dom ( A o. B ) e. _V )
8	7	adantl 277	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A o. B ) e. _V )
9		rnco 5137	. . . 4 |- ran ( A o. B ) = ran ( A |` ran B )
10		rnexg 4894	. . . . . 6 |- ( B e. C -> ran B e. _V )
11		resfunexg 5739	. . . . . 6 |- ( ( Fun A /\ ran B e. _V ) -> ( A |` ran B ) e. _V )
12	10, 11	sylan2 286	. . . . 5 |- ( ( Fun A /\ B e. C ) -> ( A |` ran B ) e. _V )
13		rnexg 4894	. . . . 5 |- ( ( A |` ran B ) e. _V -> ran ( A |` ran B ) e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( Fun A /\ B e. C ) -> ran ( A |` ran B ) e. _V )
15	9, 14	eqeltrid 2264	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A o. B ) e. _V )
16		xpexg 4742	. . 3 |- ( ( dom ( A o. B ) e. _V /\ ran ( A o. B ) e. _V ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) e. _V )
17	8, 15, 16	syl2anc 411	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) e. _V )
18		ssexg 4144	. 2 |- ( ( ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) /\ ( dom ( A o. B ) X. ran ( A o. B ) ) e. _V ) -> ( A o. B ) e. _V )
19	3, 17, 18	sylancr 414	1 |- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   _Vcvv 2739   C_ wss 3131   X. cxp 4626   dom cdm 4628   ran crn 4629   |` cres 4630   o. ccom 4632   Rel wrel 4633   Fun wfun 5212

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-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

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-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226

This theorem is referenced by:  cofunex2g  6113  ctm  7110  ctssdclemr  7113  prdsex  12723