Theorem ismrcd1 30835

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 15034), isotone (satisfies mrcss 15033), and idempotent (satisfies mrcidm 15036) 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 30836 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 3714	. . . 4 |- ( F i^i _I ) C_ F
2		dmss 5212	. . . 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 3547	. 2 |- ( ph -> dom ( F i^i _I ) C_ ~P B )
8		ssid 3518	. . . . . . 7 |- B C_ B
9		ismrcd.b	. . . . . . . 8 |- ( ph -> B e. V )
10		elpwg 4023	. . . . . . . 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 4025	. . . 4 |- ( ph -> ( F ` B ) C_ B )
15		selpw 4022	. . . . . . 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 2871	. . . . 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 3528	. . . . . 6 |- ( x = B -> ( x C_ ( F ` x ) <-> B C_ ( F ` B ) ) )
22	21	rspcva 3208	. . . . 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 3516	. . 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 3509	. . . . . . . . . . . 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 4314	. . . . . . . . . . . 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 4706	. . . . . . . . . . 11 |- U. ~P B = B
37	35, 36	syl6sseq 3545	. . . . . . . . . 10 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z C_ B )
38		intex 4612	. . . . . . . . . . . 12 |- ( z =/= (/) <-> |^| z e. _V )
39		elpwg 4023	. . . . . . . . . . . 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 1720	. . . . . . . . . 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 3499	. . . . . . . . . 10 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x e. ~P B )
50	49	elpwid 4025	. . . . . . . . 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 3520	. . . . . . . . . . 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 3526	. . . . . . . . . 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 3194	. . . . . . . 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 3499	. . . . . . . 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 3535	. . . . . 6 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( F ` |^| z ) C_ x )
68	67	ralrimiva 2871	. . . . 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 3528	. . . . . 6 |- ( x = |^| z -> ( x C_ ( F ` x ) <-> |^| z C_ ( F ` |^| z ) ) )
75	74	rspcva 3208	. . . . 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 3516	. . 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 15019	1 |- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   A.wal 1393   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251   |^|cint 4288   _I cid 4799   dom cdm 5008   Fn wfn 5589   -->wf 5590   ` cfv 5594  Moorecmre 14999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 15003

This theorem is referenced by:  ismrcd2  30836  istopclsd  30837  ismrc  30838