Theorem cofunexg 6273

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 5234	. . 3 |- Rel ( A o. B )
2		relssdmrn 5256	. . 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 5001	. . . . 5 |- dom ( A o. B ) C_ dom B
5		dmexg 4995	. . . . 5 |- ( B e. C -> dom B e. _V )
6		ssexg 4227	. . . . 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 5242	. . . 4 |- ran ( A o. B ) = ran ( A |` ran B )
10		rnexg 4996	. . . . . 6 |- ( B e. C -> ran B e. _V )
11		resfunexg 5875	. . . . . 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 4996	. . . . 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 2317	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A o. B ) e. _V )
16		xpexg 4839	. . 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 4227	. 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 2201   _Vcvv 2801   C_ wss 3199   X. cxp 4722   dom cdm 4724   ran crn 4725   |` cres 4726   o. ccom 4728   Rel wrel 4729   Fun wfun 5319

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333

This theorem is referenced by:  cofunex2g  6274  ctm  7310  ctssdclemr  7313  prdsex  13372  prdsval  13376  prdsbaslemss  13377