Theorem cofunexg 6652

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 5445	. . 3 |- Rel ( A o. B )
2		relssdmrn 5467	. . 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 5208	. . . . 5 |- dom ( A o. B ) C_ dom B
5		dmexg 6620	. . . . 5 |- ( B e. C -> dom B e. _V )
6		ssexg 4547	. . . . 5 |- ( ( dom ( A o. B ) C_ dom B /\ dom B e. _V ) -> dom ( A o. B ) e. _V )
7	4, 5, 6	sylancr 663	. . . 4 |- ( B e. C -> dom ( A o. B ) e. _V )
8	7	adantl 466	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A o. B ) e. _V )
9		rnco 5453	. . . 4 |- ran ( A o. B ) = ran ( A |` ran B )
10		rnexg 6621	. . . . . 6 |- ( B e. C -> ran B e. _V )
11		resfunexg 6053	. . . . . 6 |- ( ( Fun A /\ ran B e. _V ) -> ( A |` ran B ) e. _V )
12	10, 11	sylan2 474	. . . . 5 |- ( ( Fun A /\ B e. C ) -> ( A |` ran B ) e. _V )
13		rnexg 6621	. . . . 5 |- ( ( A |` ran B ) e. _V -> ran ( A |` ran B ) e. _V )
14	12, 13	syl 16	. . . 4 |- ( ( Fun A /\ B e. C ) -> ran ( A |` ran B ) e. _V )
15	9, 14	syl5eqel 2546	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A o. B ) e. _V )
16		xpexg 6618	. . 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 661	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) e. _V )
18		ssexg 4547	. 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 663	1 |- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1758   _Vcvv 3078   C_ wss 3437   X. cxp 4947   dom cdm 4949   ran crn 4950   |` cres 4951   o. ccom 4953   Rel wrel 4954   Fun wfun 5521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535

This theorem is referenced by:  cofunex2g  6653  fin1a2lem7  8687  revco  12581  ccatco  12582  lswco  12585  isoval  14823  bcthlem4  20971  sseqval  26916  sinccvglem  27462