Theorem sscpwex 15221

Description: An analogue of pwex 4548 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 6224	. 2 |- ( ~P U. ran J ^pm dom J ) e. _V
2		brssc 15220	. . . 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 455	. . . . . . . . . 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 3037	. . . . . . . . . . 11 |- t e. _V
5	4, 4	xpex 6503	. . . . . . . . . 10 |- ( t X. t ) e. _V
6		fnex 6040	. . . . . . . . . 10 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	3, 5, 6	sylancl 660	. . . . . . . . 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 6631	. . . . . . . . 9 |- ( J e. _V -> ran J e. _V )
9		uniexg 6496	. . . . . . . . 9 |- ( ran J e. _V -> U. ran J e. _V )
10		pwexg 4549	. . . . . . . . 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 5588	. . . . . . . . . 10 |- ( J Fn ( t X. t ) -> dom J = ( t X. t ) )
13	12	adantr 463	. . . . . . . . 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 2478	. . . . . . . 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 7401	. . . . . . . . . . 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 5797	. . . . . . . . . . . . 13 |- ( J ` x ) C_ U. ran J
17		sspwb 4611	. . . . . . . . . . . . 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 2745	. . . . . . . . . 10 |- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J
21		simprr 755	. . . . . . . . . 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 3415	. . . . . . . . 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 3037	. . . . . . . . . 10 |- h e. _V
24	23	elixpconst 7396	. . . . . . . . 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 3936	. . . . . . . . . . 11 |- ( s e. ~P t -> s C_ t )
27	26	ad2antrl 725	. . . . . . . . . 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 5021	. . . . . . . . . 10 |- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) )
29	27, 27, 28	syl2anc 659	. . . . . . . . 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 3454	. . . . . . . 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 7355	. . . . . . . 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 1227	. . . . . . 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 2875	. . . . . 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 427	. . . . 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 1730	. . . 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 3489	. 2 |- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J )
38	1, 37	ssexi 4510	1 |- { h | h C_cat J } e. _V

Syntax hints:   /\ wa 367   = wceq 1399   E.wex 1620   e. wcel 1826   {cab 2367   E.wrex 2733   _Vcvv 3034   C_ wss 3389   ~Pcpw 3927   U.cuni 4163   class class class wbr 4367   X. cxp 4911   dom cdm 4913   ran crn 4914   Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   ^pm cpm 7339   X_cixp 7388   C__cat cssc 15213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-pm 7341  df-ixp 7389  df-ssc 15216

This theorem is referenced by:  issubc  15241