Theorem ismrcd1 30558

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 14889), isotone (satisfies mrcss 14888), and idempotent (satisfies mrcidm 14891) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 30559 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
ismrcd.b	|- ( ph -> B e. V )
ismrcd.f	|- ( ph -> F : ~P B --> ~P B )
ismrcd.e	|- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )
ismrcd.m	|- ( ( ph /\ x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) )
ismrcd.i	|- ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) )

Assertion

Ref	Expression
ismrcd1	|- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )

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

Proof of Theorem ismrcd1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3723	. . . 4 |- ( F i^i _I ) C_ F
2		dmss 5208	. . . 4 |- ( ( F i^i _I ) C_ F -> dom ( F i^i _I ) C_ dom F )
3	1, 2	ax-mp 5	. . 3 |- dom ( F i^i _I ) C_ dom F
4		ismrcd.f	. . . 4 |- ( ph -> F : ~P B --> ~P B )
5		fdm 5741	. . . 4 |- ( F : ~P B --> ~P B -> dom F = ~P B )
6	4, 5	syl 16	. . 3 |- ( ph -> dom F = ~P B )
7	3, 6	syl5sseq 3557	. 2 |- ( ph -> dom ( F i^i _I ) C_ ~P B )
8		ssid 3528	. . . . . . 7 |- B C_ B
9		ismrcd.b	. . . . . . . 8 |- ( ph -> B e. V )
10		elpwg 4024	. . . . . . . 8 |- ( B e. V -> ( B e. ~P B <-> B C_ B ) )
11	9, 10	syl 16	. . . . . . 7 |- ( ph -> ( B e. ~P B <-> B C_ B ) )
12	8, 11	mpbiri 233	. . . . . 6 |- ( ph -> B e. ~P B )
13	4, 12	ffvelrnd 6033	. . . . 5 |- ( ph -> ( F ` B ) e. ~P B )
14	13	elpwid 4026	. . . 4 |- ( ph -> ( F ` B ) C_ B )
15		selpw 4023	. . . . . . 7 |- ( x e. ~P B <-> x C_ B )
16		ismrcd.e	. . . . . . 7 |- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )
17	15, 16	sylan2b 475	. . . . . 6 |- ( ( ph /\ x e. ~P B ) -> x C_ ( F ` x ) )
18	17	ralrimiva 2881	. . . . 5 |- ( ph -> A. x e. ~P B x C_ ( F ` x ) )
19		id 22	. . . . . . 7 |- ( x = B -> x = B )
20		fveq2 5872	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
21	19, 20	sseq12d 3538	. . . . . 6 |- ( x = B -> ( x C_ ( F ` x ) <-> B C_ ( F ` B ) ) )
22	21	rspcva 3217	. . . . 5 |- ( ( B e. ~P B /\ A. x e. ~P B x C_ ( F ` x ) ) -> B C_ ( F ` B ) )
23	12, 18, 22	syl2anc 661	. . . 4 |- ( ph -> B C_ ( F ` B ) )
24	14, 23	eqssd 3526	. . 3 |- ( ph -> ( F ` B ) = B )
25		ffn 5737	. . . . 5 |- ( F : ~P B --> ~P B -> F Fn ~P B )
26	4, 25	syl 16	. . . 4 |- ( ph -> F Fn ~P B )
27		fnelfp 6100	. . . 4 |- ( ( F Fn ~P B /\ B e. ~P B ) -> ( B e. dom ( F i^i _I ) <-> ( F ` B ) = B ) )
28	26, 12, 27	syl2anc 661	. . 3 |- ( ph -> ( B e. dom ( F i^i _I ) <-> ( F ` B ) = B ) )
29	24, 28	mpbird 232	. 2 |- ( ph -> B e. dom ( F i^i _I ) )
30		simp2 997	. . . . . . . . . . . . 13 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z C_ dom ( F i^i _I ) )
31	7	3ad2ant1 1017	. . . . . . . . . . . . 13 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> dom ( F i^i _I ) C_ ~P B )
32	30, 31	sstrd 3519	. . . . . . . . . . . 12 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z C_ ~P B )
33		simp3 998	. . . . . . . . . . . 12 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z =/= (/) )
34		intssuni2 4313	. . . . . . . . . . . 12 |- ( ( z C_ ~P B /\ z =/= (/) ) -> |^| z C_ U. ~P B )
35	32, 33, 34	syl2anc 661	. . . . . . . . . . 11 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z C_ U. ~P B )
36		unipw 4703	. . . . . . . . . . 11 |- U. ~P B = B
37	35, 36	syl6sseq 3555	. . . . . . . . . 10 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z C_ B )
38		intex 4609	. . . . . . . . . . . 12 |- ( z =/= (/) <-> |^| z e. _V )
39		elpwg 4024	. . . . . . . . . . . 12 |- ( |^| z e. _V -> ( |^| z e. ~P B <-> |^| z C_ B ) )
40	38, 39	sylbi 195	. . . . . . . . . . 11 |- ( z =/= (/) -> ( |^| z e. ~P B <-> |^| z C_ B ) )
41	40	3ad2ant3 1019	. . . . . . . . . 10 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> ( |^| z e. ~P B <-> |^| z C_ B ) )
42	37, 41	mpbird 232	. . . . . . . . 9 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z e. ~P B )
43	42	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> |^| z e. ~P B )
44		ismrcd.m	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) )
45	44	3expib 1199	. . . . . . . . . . 11 |- ( ph -> ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
46	45	alrimiv 1695	. . . . . . . . . 10 |- ( ph -> A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
47	46	3ad2ant1 1017	. . . . . . . . 9 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
48	47	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
49	32	sselda 3509	. . . . . . . . . 10 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x e. ~P B )
50	49	elpwid 4026	. . . . . . . . 9 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x C_ B )
51		intss1 4303	. . . . . . . . . 10 |- ( x e. z -> |^| z C_ x )
52	51	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> |^| z C_ x )
53	50, 52	jca 532	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( x C_ B /\ |^| z C_ x ) )
54		sseq1 3530	. . . . . . . . . . 11 |- ( y = |^| z -> ( y C_ x <-> |^| z C_ x ) )
55	54	anbi2d 703	. . . . . . . . . 10 |- ( y = |^| z -> ( ( x C_ B /\ y C_ x ) <-> ( x C_ B /\ |^| z C_ x ) ) )
56		fveq2 5872	. . . . . . . . . . 11 |- ( y = |^| z -> ( F ` y ) = ( F ` |^| z ) )
57	56	sseq1d 3536	. . . . . . . . . 10 |- ( y = |^| z -> ( ( F ` y ) C_ ( F ` x ) <-> ( F ` |^| z ) C_ ( F ` x ) ) )
58	55, 57	imbi12d 320	. . . . . . . . 9 |- ( y = |^| z -> ( ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) <-> ( ( x C_ B /\ |^| z C_ x ) -> ( F ` |^| z ) C_ ( F ` x ) ) ) )
59	58	spcgv 3203	. . . . . . . 8 |- ( |^| z e. ~P B -> ( A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) -> ( ( x C_ B /\ |^| z C_ x ) -> ( F ` |^| z ) C_ ( F ` x ) ) ) )
60	43, 48, 53, 59	syl3c 61	. . . . . . 7 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( F ` |^| z ) C_ ( F ` x ) )
61	30	sselda 3509	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x e. dom ( F i^i _I ) )
62	26	3ad2ant1 1017	. . . . . . . . . 10 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> F Fn ~P B )
63	62	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> F Fn ~P B )
64		fnelfp 6100	. . . . . . . . 9 |- ( ( F Fn ~P B /\ x e. ~P B ) -> ( x e. dom ( F i^i _I ) <-> ( F ` x ) = x ) )
65	63, 49, 64	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( x e. dom ( F i^i _I ) <-> ( F ` x ) = x ) )
66	61, 65	mpbid 210	. . . . . . 7 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( F ` x ) = x )
67	60, 66	sseqtrd 3545	. . . . . 6 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( F ` |^| z ) C_ x )
68	67	ralrimiva 2881	. . . . 5 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> A. x e. z ( F ` |^| z ) C_ x )
69		ssint 4304	. . . . 5 |- ( ( F ` |^| z ) C_ |^| z <-> A. x e. z ( F ` |^| z ) C_ x )
70	68, 69	sylibr 212	. . . 4 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> ( F ` |^| z ) C_ |^| z )
71	18	3ad2ant1 1017	. . . . 5 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> A. x e. ~P B x C_ ( F ` x ) )
72		id 22	. . . . . . 7 |- ( x = |^| z -> x = |^| z )
73		fveq2 5872	. . . . . . 7 |- ( x = |^| z -> ( F ` x ) = ( F ` |^| z ) )
74	72, 73	sseq12d 3538	. . . . . 6 |- ( x = |^| z -> ( x C_ ( F ` x ) <-> |^| z C_ ( F ` |^| z ) ) )
75	74	rspcva 3217	. . . . 5 |- ( ( |^| z e. ~P B /\ A. x e. ~P B x C_ ( F ` x ) ) -> |^| z C_ ( F ` |^| z ) )
76	42, 71, 75	syl2anc 661	. . . 4 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z C_ ( F ` |^| z ) )
77	70, 76	eqssd 3526	. . 3 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> ( F ` |^| z ) = |^| z )
78		fnelfp 6100	. . . 4 |- ( ( F Fn ~P B /\ |^| z e. ~P B ) -> ( |^| z e. dom ( F i^i _I ) <-> ( F ` |^| z ) = |^| z ) )
79	62, 42, 78	syl2anc 661	. . 3 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> ( |^| z e. dom ( F i^i _I ) <-> ( F ` |^| z ) = |^| z ) )
80	77, 79	mpbird 232	. 2 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z e. dom ( F i^i _I ) )
81	7, 29, 80	ismred 14874	1 |- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   A.wal 1377   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   _Vcvv 3118   i^i cin 3480   C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   |^|cint 4288   _I cid 4796   dom cdm 5005   Fn wfn 5589   -->wf 5590   ` cfv 5594  Moorecmre 14854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 14858

This theorem is referenced by:  ismrcd2  30559  istopclsd  30560  ismrc  30561