Theorem cofunexg 6311

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 5266	. . 3 |- Rel ( A o. B )
2		relssdmrn 5288	. . 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 5032	. . . . 5 |- dom ( A o. B ) C_ dom B
5		dmexg 5026	. . . . 5 |- ( B e. C -> dom B e. _V )
6		ssexg 4254	. . . . 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 5274	. . . 4 |- ran ( A o. B ) = ran ( A |` ran B )
10		rnexg 5027	. . . . . 6 |- ( B e. C -> ran B e. _V )
11		resfunexg 5910	. . . . . 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 5027	. . . . 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 2321	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A o. B ) e. _V )
16		xpexg 4869	. . 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 4254	. 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 2205   _Vcvv 2815   C_ wss 3214   X. cxp 4752   dom cdm 4754   ran crn 4755   |` cres 4756   o. ccom 4758   Rel wrel 4759   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  cofunex2g  6312  ctm  7413  ctssdclemr  7416  prdsex  14114  prdsval  14115  prdsbaslemss  14116