Theorem sscpwex 15720

Description: An analogue of pwex 4586 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 6318	. 2 |- ( ~P U. ran J ^pm dom J ) e. _V
2		brssc 15719	. . . 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 459	. . . . . . . . . 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 3048	. . . . . . . . . . 11 |- t e. _V
5	4, 4	xpex 6595	. . . . . . . . . 10 |- ( t X. t ) e. _V
6		fnex 6132	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	3, 5, 6	sylancl 668	. . . . . . . . 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 6725	. . . . . . . . 9 |- ( J e. _V -> ran J e. _V )
9		uniexg 6588	. . . . . . . . 9 |- ( ran J e. _V -> U. ran J e. _V )
10		pwexg 4587	. . . . . . . . 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 5675	. . . . . . . . . 10 |- ( J Fn ( t X. t ) -> dom J = ( t X. t ) )
13	12	adantr 467	. . . . . . . . 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 2537	. . . . . . . 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 7535	. . . . . . . . . . 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 5888	. . . . . . . . . . . . 13 |- ( J ` x ) C_ U. ran J
17		sspwb 4649	. . . . . . . . . . . . 13 |- ( ( J ` x ) C_ U. ran J <-> ~P ( J ` x ) C_ ~P U. ran J )
18	16, 17	mpbi 212	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 |- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J
21		simprr 766	. . . . . . . . . 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 3430	. . . . . . . . 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 3048	. . . . . . . . . 10 |- h e. _V
24	23	elixpconst 7530	. . . . . . . . 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 200	. . . . . . . 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 3960	. . . . . . . . . . 11 |- ( s e. ~P t -> s C_ t )
27	26	ad2antrl 734	. . . . . . . . . 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 4940	. . . . . . . . . 10 |- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) )
29	27, 27, 28	syl2anc 667	. . . . . . . . 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 3469	. . . . . . . 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 7489	. . . . . . . 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 1269	. . . . . . 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 2880	. . . . . 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 431	. . . . 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 1776	. . . 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 199	. . 3 |- ( h C_cat J -> h e. ( ~P U. ran J ^pm dom J ) )
37	36	abssi 3504	. 2 |- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J )
38	1, 37	ssexi 4548	1 |- { h | h C_cat J } e. _V

Syntax hints:   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045   C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402   X. cxp 4832   dom cdm 4834   ran crn 4835   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   ^pm cpm 7473   X_cixp 7522   C__cat cssc 15712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-pm 7475  df-ixp 7523  df-ssc 15715

This theorem is referenced by:  issubc  15740