Theorem sscpwex 15671

Description: An analogue of pwex 4608 for the subcategory subset relation: The collection of subcategory subsets of a given set J is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
sscpwex	|- { h | h C_cat J } e. _V

Distinct variable group:   h, J

Proof of Theorem sscpwex

Dummy variables s t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6333	. 2 |- ( ~P U. ran J ^pm dom J ) e. _V
2		brssc 15670	. . . 4 |- ( h C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )
3		simpl 458	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> J Fn ( t X. t ) )
4		vex 3090	. . . . . . . . . . 11 |- t e. _V
5	4, 4	xpex 6609	. . . . . . . . . 10 |- ( t X. t ) e. _V
6		fnex 6147	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	3, 5, 6	sylancl 666	. . . . . . . . 9 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> J e. _V )
8		rnexg 6739	. . . . . . . . 9 |- ( J e. _V -> ran J e. _V )
9		uniexg 6602	. . . . . . . . 9 |- ( ran J e. _V -> U. ran J e. _V )
10		pwexg 4609	. . . . . . . . 9 |- ( U. ran J e. _V -> ~P U. ran J e. _V )
11	7, 8, 9, 10	4syl 19	. . . . . . . 8 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ~P U. ran J e. _V )
12		fndm 5693	. . . . . . . . . 10 |- ( J Fn ( t X. t ) -> dom J = ( t X. t ) )
13	12	adantr 466	. . . . . . . . 9 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> dom J = ( t X. t ) )
14	13, 5	syl6eqel 2525	. . . . . . . 8 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> dom J e. _V )
15		ss2ixp 7543	. . . . . . . . . . 11 |- ( A. x e. ( s X. s ) ~P ( J ` x ) C_ ~P U. ran J -> X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J )
16		fvssunirn 5904	. . . . . . . . . . . . 13 |- ( J ` x ) C_ U. ran J
17		sspwb 4671	. . . . . . . . . . . . 13 |- ( ( J ` x ) C_ U. ran J <-> ~P ( J ` x ) C_ ~P U. ran J )
18	16, 17	mpbi 211	. . . . . . . . . . . 12 |- ~P ( J ` x ) C_ ~P U. ran J
19	18	a1i 11	. . . . . . . . . . 11 |- ( x e. ( s X. s ) -> ~P ( J ` x ) C_ ~P U. ran J )
20	15, 19	mprg 2795	. . . . . . . . . 10 |- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J
21		simprr 764	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. X_ x e. ( s X. s ) ~P ( J ` x ) )
22	20, 21	sseldi 3468	. . . . . . . . 9 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. X_ x e. ( s X. s ) ~P U. ran J )
23		vex 3090	. . . . . . . . . 10 |- h e. _V
24	23	elixpconst 7538	. . . . . . . . 9 |- ( h e. X_ x e. ( s X. s ) ~P U. ran J <-> h : ( s X. s ) --> ~P U. ran J )
25	22, 24	sylib 199	. . . . . . . 8 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h : ( s X. s ) --> ~P U. ran J )
26		elpwi 3994	. . . . . . . . . . 11 |- ( s e. ~P t -> s C_ t )
27	26	ad2antrl 732	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> s C_ t )
28		xpss12 4960	. . . . . . . . . 10 |- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) )
29	27, 27, 28	syl2anc 665	. . . . . . . . 9 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ( s X. s ) C_ ( t X. t ) )
30	29, 13	sseqtr4d 3507	. . . . . . . 8 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ( s X. s ) C_ dom J )
31		elpm2r 7497	. . . . . . . 8 |- ( ( ( ~P U. ran J e. _V /\ dom J e. _V ) /\ ( h : ( s X. s ) --> ~P U. ran J /\ ( s X. s ) C_ dom J ) ) -> h e. ( ~P U. ran J ^pm dom J ) )
32	11, 14, 25, 30, 31	syl22anc 1265	. . . . . . 7 |- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. ( ~P U. ran J ^pm dom J ) )
33	32	rexlimdvaa 2925	. . . . . 6 |- ( J Fn ( t X. t ) -> ( E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) -> h e. ( ~P U. ran J ^pm dom J ) ) )
34	33	imp 430	. . . . 5 |- ( ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> h e. ( ~P U. ran J ^pm dom J ) )
35	34	exlimiv 1769	. . . 4 |- ( E. t ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> h e. ( ~P U. ran J ^pm dom J ) )
36	2, 35	sylbi 198	. . 3 |- ( h C_cat J -> h e. ( ~P U. ran J ^pm dom J ) )
37	36	abssi 3542	. 2 |- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J )
38	1, 37	ssexi 4570	1 |- { h | h C_cat J } e. _V

Syntax hints:   /\ wa 370   = wceq 1437   E.wex 1659   e. wcel 1870   {cab 2414   E.wrex 2783   _Vcvv 3087   C_ wss 3442   ~Pcpw 3985   U.cuni 4222   class class class wbr 4426   X. cxp 4852   dom cdm 4854   ran crn 4855   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   ^pm cpm 7481   X_cixp 7530   C__cat cssc 15663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-pm 7483  df-ixp 7531  df-ssc 15666

This theorem is referenced by:  issubc  15691