Theorem ismrc 37264

Description: A function is a Moore closure operator iff it satisfies mrcssid 16277, mrcss 16276, and mrcidm 16279. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
ismrc	|- ( F e. (mrCls " (Moore ` B ) ) <-> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )

Distinct variable groups:   x, F, y   x, B, y

Proof of Theorem ismrc

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmrc 16267	. . . . 5 |- mrCls Fn U. ran Moore
2		fnfun 5988	. . . . 5 |- (mrCls Fn U. ran Moore -> Fun mrCls )
3	1, 2	ax-mp 5	. . . 4 |- Fun mrCls
4		fvelima 6248	. . . 4 |- ( ( Fun mrCls /\ F e. (mrCls " (Moore ` B ) ) ) -> E. z e. (Moore ` B ) (mrCls ` z ) = F )
5	3, 4	mpan 706	. . 3 |- ( F e. (mrCls " (Moore ` B ) ) -> E. z e. (Moore ` B ) (mrCls ` z ) = F )
6		elfvex 6221	. . . . . 6 |- ( z e. (Moore ` B ) -> B e. _V )
7		eqid 2622	. . . . . . . 8 |- (mrCls ` z ) = (mrCls ` z )
8	7	mrcf 16269	. . . . . . 7 |- ( z e. (Moore ` B ) -> (mrCls ` z ) : ~P B --> z )
9		mresspw 16252	. . . . . . 7 |- ( z e. (Moore ` B ) -> z C_ ~P B )
10	8, 9	fssd 6057	. . . . . 6 |- ( z e. (Moore ` B ) -> (mrCls ` z ) : ~P B --> ~P B )
11	7	mrcssid 16277	. . . . . . . . . 10 |- ( ( z e. (Moore ` B ) /\ x C_ B ) -> x C_ ( (mrCls ` z ) ` x ) )
12	11	adantrr 753	. . . . . . . . 9 |- ( ( z e. (Moore ` B ) /\ ( x C_ B /\ y C_ x ) ) -> x C_ ( (mrCls ` z ) ` x ) )
13	7	mrcss 16276	. . . . . . . . . . 11 |- ( ( z e. (Moore ` B ) /\ y C_ x /\ x C_ B ) -> ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) )
14	13	3expb 1266	. . . . . . . . . 10 |- ( ( z e. (Moore ` B ) /\ ( y C_ x /\ x C_ B ) ) -> ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) )
15	14	ancom2s 844	. . . . . . . . 9 |- ( ( z e. (Moore ` B ) /\ ( x C_ B /\ y C_ x ) ) -> ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) )
16	7	mrcidm 16279	. . . . . . . . . 10 |- ( ( z e. (Moore ` B ) /\ x C_ B ) -> ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) )
17	16	adantrr 753	. . . . . . . . 9 |- ( ( z e. (Moore ` B ) /\ ( x C_ B /\ y C_ x ) ) -> ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) )
18	12, 15, 17	3jca 1242	. . . . . . . 8 |- ( ( z e. (Moore ` B ) /\ ( x C_ B /\ y C_ x ) ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) )
19	18	ex 450	. . . . . . 7 |- ( z e. (Moore ` B ) -> ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) )
20	19	alrimivv 1856	. . . . . 6 |- ( z e. (Moore ` B ) -> A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) )
21	6, 10, 20	3jca 1242	. . . . 5 |- ( z e. (Moore ` B ) -> ( B e. _V /\ (mrCls ` z ) : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) ) )
22		feq1 6026	. . . . . 6 |- ( (mrCls ` z ) = F -> ( (mrCls ` z ) : ~P B --> ~P B <-> F : ~P B --> ~P B ) )
23		fveq1 6190	. . . . . . . . . 10 |- ( (mrCls ` z ) = F -> ( (mrCls ` z ) ` x ) = ( F ` x ) )
24	23	sseq2d 3633	. . . . . . . . 9 |- ( (mrCls ` z ) = F -> ( x C_ ( (mrCls ` z ) ` x ) <-> x C_ ( F ` x ) ) )
25		fveq1 6190	. . . . . . . . . 10 |- ( (mrCls ` z ) = F -> ( (mrCls ` z ) ` y ) = ( F ` y ) )
26	25, 23	sseq12d 3634	. . . . . . . . 9 |- ( (mrCls ` z ) = F -> ( ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) <-> ( F ` y ) C_ ( F ` x ) ) )
27		id 22	. . . . . . . . . . 11 |- ( (mrCls ` z ) = F -> (mrCls ` z ) = F )
28	27, 23	fveq12d 6197	. . . . . . . . . 10 |- ( (mrCls ` z ) = F -> ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( F ` ( F ` x ) ) )
29	28, 23	eqeq12d 2637	. . . . . . . . 9 |- ( (mrCls ` z ) = F -> ( ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) <-> ( F ` ( F ` x ) ) = ( F ` x ) ) )
30	24, 26, 29	3anbi123d 1399	. . . . . . . 8 |- ( (mrCls ` z ) = F -> ( ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) <-> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) )
31	30	imbi2d 330	. . . . . . 7 |- ( (mrCls ` z ) = F -> ( ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) <-> ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )
32	31	2albidv 1851	. . . . . 6 |- ( (mrCls ` z ) = F -> ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) <-> A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )
33	22, 32	3anbi23d 1402	. . . . 5 |- ( (mrCls ` z ) = F -> ( ( B e. _V /\ (mrCls ` z ) : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` y ) C_ ( (mrCls ` z ) ` x ) /\ ( (mrCls ` z ) ` ( (mrCls ` z ) ` x ) ) = ( (mrCls ` z ) ` x ) ) ) ) <-> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) ) )
34	21, 33	syl5ibcom 235	. . . 4 |- ( z e. (Moore ` B ) -> ( (mrCls ` z ) = F -> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) ) )
35	34	rexlimiv 3027	. . 3 |- ( E. z e. (Moore ` B ) (mrCls ` z ) = F -> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )
36	5, 35	syl 17	. 2 |- ( F e. (mrCls " (Moore ` B ) ) -> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )
37		simp1 1061	. . . 4 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> B e. _V )
38		simp2 1062	. . . 4 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> F : ~P B --> ~P B )
39		ssid 3624	. . . . . . 7 |- z C_ z
40		3simpb 1059	. . . . . . . . . . 11 |- ( ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) )
41	40	imim2i 16	. . . . . . . . . 10 |- ( ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) )
42	41	2alimi 1740	. . . . . . . . 9 |- ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) )
43		vex 3203	. . . . . . . . . 10 |- z e. _V
44		sseq1 3626	. . . . . . . . . . . . . 14 |- ( x = z -> ( x C_ B <-> z C_ B ) )
45	44	adantr 481	. . . . . . . . . . . . 13 |- ( ( x = z /\ y = z ) -> ( x C_ B <-> z C_ B ) )
46		sseq12 3628	. . . . . . . . . . . . . 14 |- ( ( y = z /\ x = z ) -> ( y C_ x <-> z C_ z ) )
47	46	ancoms 469	. . . . . . . . . . . . 13 |- ( ( x = z /\ y = z ) -> ( y C_ x <-> z C_ z ) )
48	45, 47	anbi12d 747	. . . . . . . . . . . 12 |- ( ( x = z /\ y = z ) -> ( ( x C_ B /\ y C_ x ) <-> ( z C_ B /\ z C_ z ) ) )
49		id 22	. . . . . . . . . . . . . . 15 |- ( x = z -> x = z )
50		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = z -> ( F ` x ) = ( F ` z ) )
51	49, 50	sseq12d 3634	. . . . . . . . . . . . . 14 |- ( x = z -> ( x C_ ( F ` x ) <-> z C_ ( F ` z ) ) )
52	51	adantr 481	. . . . . . . . . . . . 13 |- ( ( x = z /\ y = z ) -> ( x C_ ( F ` x ) <-> z C_ ( F ` z ) ) )
53	50	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( x = z -> ( F ` ( F ` x ) ) = ( F ` ( F ` z ) ) )
54	53, 50	eqeq12d 2637	. . . . . . . . . . . . . 14 |- ( x = z -> ( ( F ` ( F ` x ) ) = ( F ` x ) <-> ( F ` ( F ` z ) ) = ( F ` z ) ) )
55	54	adantr 481	. . . . . . . . . . . . 13 |- ( ( x = z /\ y = z ) -> ( ( F ` ( F ` x ) ) = ( F ` x ) <-> ( F ` ( F ` z ) ) = ( F ` z ) ) )
56	52, 55	anbi12d 747	. . . . . . . . . . . 12 |- ( ( x = z /\ y = z ) -> ( ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) <-> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
57	48, 56	imbi12d 334	. . . . . . . . . . 11 |- ( ( x = z /\ y = z ) -> ( ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) <-> ( ( z C_ B /\ z C_ z ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) ) )
58	57	spc2gv 3296	. . . . . . . . . 10 |- ( ( z e. _V /\ z e. _V ) -> ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> ( ( z C_ B /\ z C_ z ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) ) )
59	43, 43, 58	mp2an 708	. . . . . . . . 9 |- ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> ( ( z C_ B /\ z C_ z ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
60	42, 59	syl 17	. . . . . . . 8 |- ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> ( ( z C_ B /\ z C_ z ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
61	60	3ad2ant3 1084	. . . . . . 7 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> ( ( z C_ B /\ z C_ z ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
62	39, 61	mpan2i 713	. . . . . 6 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> ( z C_ B -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
63	62	imp 445	. . . . 5 |- ( ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) /\ z C_ B ) -> ( z C_ ( F ` z ) /\ ( F ` ( F ` z ) ) = ( F ` z ) ) )
64	63	simpld 475	. . . 4 |- ( ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) /\ z C_ B ) -> z C_ ( F ` z ) )
65		simp2 1062	. . . . . . . . 9 |- ( ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) -> ( F ` y ) C_ ( F ` x ) )
66	65	imim2i 16	. . . . . . . 8 |- ( ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
67	66	2alimi 1740	. . . . . . 7 |- ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) -> A. x A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
68	67	3ad2ant3 1084	. . . . . 6 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> A. x A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
69		vex 3203	. . . . . . 7 |- w e. _V
70	44	adantr 481	. . . . . . . . . 10 |- ( ( x = z /\ y = w ) -> ( x C_ B <-> z C_ B ) )
71		sseq12 3628	. . . . . . . . . . 11 |- ( ( y = w /\ x = z ) -> ( y C_ x <-> w C_ z ) )
72	71	ancoms 469	. . . . . . . . . 10 |- ( ( x = z /\ y = w ) -> ( y C_ x <-> w C_ z ) )
73	70, 72	anbi12d 747	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( ( x C_ B /\ y C_ x ) <-> ( z C_ B /\ w C_ z ) ) )
74		fveq2 6191	. . . . . . . . . 10 |- ( y = w -> ( F ` y ) = ( F ` w ) )
75		sseq12 3628	. . . . . . . . . 10 |- ( ( ( F ` y ) = ( F ` w ) /\ ( F ` x ) = ( F ` z ) ) -> ( ( F ` y ) C_ ( F ` x ) <-> ( F ` w ) C_ ( F ` z ) ) )
76	74, 50, 75	syl2anr 495	. . . . . . . . 9 |- ( ( x = z /\ y = w ) -> ( ( F ` y ) C_ ( F ` x ) <-> ( F ` w ) C_ ( F ` z ) ) )
77	73, 76	imbi12d 334	. . . . . . . 8 |- ( ( x = z /\ y = w ) -> ( ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) <-> ( ( z C_ B /\ w C_ z ) -> ( F ` w ) C_ ( F ` z ) ) ) )
78	77	spc2gv 3296	. . . . . . 7 |- ( ( z e. _V /\ w e. _V ) -> ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) -> ( ( z C_ B /\ w C_ z ) -> ( F ` w ) C_ ( F ` z ) ) ) )
79	43, 69, 78	mp2an 708	. . . . . 6 |- ( A. x A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) -> ( ( z C_ B /\ w C_ z ) -> ( F ` w ) C_ ( F ` z ) ) )
80	68, 79	syl 17	. . . . 5 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> ( ( z C_ B /\ w C_ z ) -> ( F ` w ) C_ ( F ` z ) ) )
81	80	3impib 1262	. . . 4 |- ( ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) /\ z C_ B /\ w C_ z ) -> ( F ` w ) C_ ( F ` z ) )
82	63	simprd 479	. . . 4 |- ( ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) /\ z C_ B ) -> ( F ` ( F ` z ) ) = ( F ` z ) )
83	37, 38, 64, 81, 82	ismrcd2 37262	. . 3 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> F = (mrCls ` dom ( F i^i _I ) ) )
84	37, 38, 64, 81, 82	ismrcd1 37261	. . . 4 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> dom ( F i^i _I ) e. (Moore ` B ) )
85		fvssunirn 6217	. . . . . 6 |- (Moore ` B ) C_ U. ran Moore
86		fndm 5990	. . . . . . 7 |- (mrCls Fn U. ran Moore -> dom mrCls = U. ran Moore )
87	1, 86	ax-mp 5	. . . . . 6 |- dom mrCls = U. ran Moore
88	85, 87	sseqtr4i 3638	. . . . 5 |- (Moore ` B ) C_ dom mrCls
89		funfvima2 6493	. . . . 5 |- ( ( Fun mrCls /\ (Moore ` B ) C_ dom mrCls ) -> ( dom ( F i^i _I ) e. (Moore ` B ) -> (mrCls ` dom ( F i^i _I ) ) e. (mrCls " (Moore ` B ) ) ) )
90	3, 88, 89	mp2an 708	. . . 4 |- ( dom ( F i^i _I ) e. (Moore ` B ) -> (mrCls ` dom ( F i^i _I ) ) e. (mrCls " (Moore ` B ) ) )
91	84, 90	syl 17	. . 3 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> (mrCls ` dom ( F i^i _I ) ) e. (mrCls " (Moore ` B ) ) )
92	83, 91	eqeltrd 2701	. 2 |- ( ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) -> F e. (mrCls " (Moore ` B ) ) )
93	36, 92	impbii 199	1 |- ( F e. (mrCls " (Moore ` B ) ) <-> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   _I cid 5023   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

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-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-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-int 4476  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-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by: (None)