Theorem cofunexg 6088

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 5109	. . 3 |- Rel ( A o. B )
2		relssdmrn 5131	. . 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 4880	. . . . 5 |- dom ( A o. B ) C_ dom B
5		dmexg 4875	. . . . 5 |- ( B e. C -> dom B e. _V )
6		ssexg 4128	. . . . 5 |- ( ( dom ( A o. B ) C_ dom B /\ dom B e. _V ) -> dom ( A o. B ) e. _V )
7	4, 5, 6	sylancr 412	. . . 4 |- ( B e. C -> dom ( A o. B ) e. _V )
8	7	adantl 275	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A o. B ) e. _V )
9		rnco 5117	. . . 4 |- ran ( A o. B ) = ran ( A |` ran B )
10		rnexg 4876	. . . . . 6 |- ( B e. C -> ran B e. _V )
11		resfunexg 5717	. . . . . 6 |- ( ( Fun A /\ ran B e. _V ) -> ( A |` ran B ) e. _V )
12	10, 11	sylan2 284	. . . . 5 |- ( ( Fun A /\ B e. C ) -> ( A |` ran B ) e. _V )
13		rnexg 4876	. . . . 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 2257	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A o. B ) e. _V )
16		xpexg 4725	. . 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 409	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) e. _V )
18		ssexg 4128	. 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 412	1 |- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   _Vcvv 2730   C_ wss 3121   X. cxp 4609   dom cdm 4611   ran crn 4612   |` cres 4613   o. ccom 4615   Rel wrel 4616   Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by:  cofunex2g  6089  ctm  7086  ctssdclemr  7089