Theorem ismrcd1 29034

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 14555), isotone (satisfies mrcss 14554), and idempotent (satisfies mrcidm 14557) 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 29035 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 3570	. . . 4 |- ( F i^i _I ) C_ F
2		dmss 5039	. . . 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 5563	. . . 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 3404	. 2 |- ( ph -> dom ( F i^i _I ) C_ ~P B )
8		ssid 3375	. . . . . . 7 |- B C_ B
9		ismrcd.b	. . . . . . . 8 |- ( ph -> B e. V )
10		elpwg 3868	. . . . . . . 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 5844	. . . . 5 |- ( ph -> ( F ` B ) e. ~P B )
14	13	elpwid 3870	. . . 4 |- ( ph -> ( F ` B ) C_ B )
15		selpw 3867	. . . . . . 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 2799	. . . . 5 |- ( ph -> A. x e. ~P B x C_ ( F ` x ) )
19		id 22	. . . . . . 7 |- ( x = B -> x = B )
20		fveq2 5691	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
21	19, 20	sseq12d 3385	. . . . . 6 |- ( x = B -> ( x C_ ( F ` x ) <-> B C_ ( F ` B ) ) )
22	21	rspcva 3071	. . . . 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 3373	. . 3 |- ( ph -> ( F ` B ) = B )
25		ffn 5559	. . . . 5 |- ( F : ~P B --> ~P B -> F Fn ~P B )
26	4, 25	syl 16	. . . 4 |- ( ph -> F Fn ~P B )
27		fnelfp 5906	. . . 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 989	. . . . . . . . . . . . 13 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z C_ dom ( F i^i _I ) )
31	7	3ad2ant1 1009	. . . . . . . . . . . . 13 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> dom ( F i^i _I ) C_ ~P B )
32	30, 31	sstrd 3366	. . . . . . . . . . . 12 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z C_ ~P B )
33		simp3 990	. . . . . . . . . . . 12 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> z =/= (/) )
34		intssuni2 4153	. . . . . . . . . . . 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 4542	. . . . . . . . . . 11 |- U. ~P B = B
37	35, 36	syl6sseq 3402	. . . . . . . . . 10 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> |^| z C_ B )
38		intex 4448	. . . . . . . . . . . 12 |- ( z =/= (/) <-> |^| z e. _V )
39		elpwg 3868	. . . . . . . . . . . 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 1011	. . . . . . . . . 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 1190	. . . . . . . . . . 11 |- ( ph -> ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
46	45	alrimiv 1685	. . . . . . . . . 10 |- ( ph -> A. y ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
47	46	3ad2ant1 1009	. . . . . . . . 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 3356	. . . . . . . . . 10 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x e. ~P B )
50	49	elpwid 3870	. . . . . . . . 9 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x C_ B )
51		intss1 4143	. . . . . . . . . 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 3377	. . . . . . . . . . 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 5691	. . . . . . . . . . 11 |- ( y = |^| z -> ( F ` y ) = ( F ` |^| z ) )
57	56	sseq1d 3383	. . . . . . . . . 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 3057	. . . . . . . 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 3356	. . . . . . . 8 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> x e. dom ( F i^i _I ) )
62	26	3ad2ant1 1009	. . . . . . . . . 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 5906	. . . . . . . . 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 3392	. . . . . 6 |- ( ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) /\ x e. z ) -> ( F ` |^| z ) C_ x )
68	67	ralrimiva 2799	. . . . 5 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> A. x e. z ( F ` |^| z ) C_ x )
69		ssint 4144	. . . . 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 1009	. . . . 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 5691	. . . . . . 7 |- ( x = |^| z -> ( F ` x ) = ( F ` |^| z ) )
74	72, 73	sseq12d 3385	. . . . . 6 |- ( x = |^| z -> ( x C_ ( F ` x ) <-> |^| z C_ ( F ` |^| z ) ) )
75	74	rspcva 3071	. . . . 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 3373	. . 3 |- ( ( ph /\ z C_ dom ( F i^i _I ) /\ z =/= (/) ) -> ( F ` |^| z ) = |^| z )
78		fnelfp 5906	. . . 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 14540	1 |- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   A.wal 1367   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   _Vcvv 2972   i^i cin 3327   C_ wss 3328   (/)c0 3637   ~Pcpw 3860   U.cuni 4091   |^|cint 4128   _I cid 4631   dom cdm 4840   Fn wfn 5413   -->wf 5414   ` cfv 5418  Moorecmre 14520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-mre 14524

This theorem is referenced by:  ismrcd2  29035  istopclsd  29036  ismrc  29037