Theorem sscpwex 14728

Description: An analogue of pwex 4475 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 6116	. 2 |- ( ~P U. ran J ^pm dom J ) e. _V
2		brssc 14727	. . . 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 457	. . . . . . . . . 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 2975	. . . . . . . . . . 11 |- t e. _V
5	4, 4	xpex 6508	. . . . . . . . . 10 |- ( t X. t ) e. _V
6		fnex 5944	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	3, 5, 6	sylancl 662	. . . . . . . . 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 6510	. . . . . . . . 9 |- ( J e. _V -> ran J e. _V )
9		uniexg 6377	. . . . . . . . 9 |- ( ran J e. _V -> U. ran J e. _V )
10		pwexg 4476	. . . . . . . . 9 |- ( U. ran J e. _V -> ~P U. ran J e. _V )
11	7, 8, 9, 10	4syl 21	. . . . . . . 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 5510	. . . . . . . . . 10 |- ( J Fn ( t X. t ) -> dom J = ( t X. t ) )
13	12	adantr 465	. . . . . . . . 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 2531	. . . . . . . 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 7276	. . . . . . . . . . 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 5713	. . . . . . . . . . . . 13 |- ( J ` x ) C_ U. ran J
17		sspwb 4541	. . . . . . . . . . . . 13 |- ( ( J ` x ) C_ U. ran J <-> ~P ( J ` x ) C_ ~P U. ran J )
18	16, 17	mpbi 208	. . . . . . . . . . . 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 2785	. . . . . . . . . 10 |- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J
21		simprr 756	. . . . . . . . . 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 3354	. . . . . . . . 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 2975	. . . . . . . . . 10 |- h e. _V
24	23	elixpconst 7271	. . . . . . . . 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 196	. . . . . . . 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 3869	. . . . . . . . . . 11 |- ( s e. ~P t -> s C_ t )
27	26	ad2antrl 727	. . . . . . . . . 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 4945	. . . . . . . . . 10 |- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) )
29	27, 27, 28	syl2anc 661	. . . . . . . . 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 3393	. . . . . . . 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 7230	. . . . . . . 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 1219	. . . . . . 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 2842	. . . . . 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 429	. . . . 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 1688	. . . 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 195	. . 3 |- ( h C_cat J -> h e. ( ~P U. ran J ^pm dom J ) )
37	36	abssi 3427	. 2 |- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J )
38	1, 37	ssexi 4437	1 |- { h | h C_cat J } e. _V

Syntax hints:   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   E.wrex 2716   _Vcvv 2972   C_ wss 3328   ~Pcpw 3860   U.cuni 4091   class class class wbr 4292   X. cxp 4838   dom cdm 4840   ran crn 4841   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   ^pm cpm 7215   X_cixp 7263   C__cat cssc 14720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-pm 7217  df-ixp 7264  df-ssc 14723

This theorem is referenced by:  issubc  14748