Theorem sscpwex 16475

Description: An analogue of pwex 4848 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 6678	. 2 |- ( ~P U. ran J ^pm dom J ) e. _V
2		brssc 16474	. . . 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 473	. . . . . . . . . 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 3203	. . . . . . . . . . 11 |- t e. _V
5	4, 4	xpex 6962	. . . . . . . . . 10 |- ( t X. t ) e. _V
6		fnex 6481	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	3, 5, 6	sylancl 694	. . . . . . . . 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 7098	. . . . . . . . 9 |- ( J e. _V -> ran J e. _V )
9		uniexg 6955	. . . . . . . . 9 |- ( ran J e. _V -> U. ran J e. _V )
10		pwexg 4850	. . . . . . . . 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 5990	. . . . . . . . . 10 |- ( J Fn ( t X. t ) -> dom J = ( t X. t ) )
13	12	adantr 481	. . . . . . . . 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 2709	. . . . . . . 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 7921	. . . . . . . . . . 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 6217	. . . . . . . . . . . . 13 |- ( J ` x ) C_ U. ran J
17		sspwb 4917	. . . . . . . . . . . . 13 |- ( ( J ` x ) C_ U. ran J <-> ~P ( J ` x ) C_ ~P U. ran J )
18	16, 17	mpbi 220	. . . . . . . . . . . 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 2926	. . . . . . . . . 10 |- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J
21		simprr 796	. . . . . . . . . 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 3601	. . . . . . . . 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 3203	. . . . . . . . . 10 |- h e. _V
24	23	elixpconst 7916	. . . . . . . . 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 208	. . . . . . . 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 4168	. . . . . . . . . . 11 |- ( s e. ~P t -> s C_ t )
27	26	ad2antrl 764	. . . . . . . . . 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 5225	. . . . . . . . . 10 |- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) )
29	27, 27, 28	syl2anc 693	. . . . . . . . 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 3642	. . . . . . . 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 7875	. . . . . . . 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 1327	. . . . . . 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 3032	. . . . . 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 445	. . . . 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 1858	. . . 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 207	. . 3 |- ( h C_cat J -> h e. ( ~P U. ran J ^pm dom J ) )
37	36	abssi 3677	. 2 |- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J )
38	1, 37	ssexi 4803	1 |- { h | h C_cat J } e. _V

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   X_cixp 7908   C__cat cssc 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-ixp 7909  df-ssc 16470

This theorem is referenced by:  issubc  16495