Theorem ismrcd2 37262

Description: Second half of ismrcd1 37261. (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
ismrcd2	|- ( ph -> F = (mrCls ` dom ( F i^i _I ) ) )

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

Proof of Theorem ismrcd2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismrcd.f	. . 3 |- ( ph -> F : ~P B --> ~P B )
2		ffn 6045	. . 3 |- ( F : ~P B --> ~P B -> F Fn ~P B )
3	1, 2	syl 17	. 2 |- ( ph -> F Fn ~P B )
4		ismrcd.b	. . . 4 |- ( ph -> B e. V )
5		ismrcd.e	. . . 4 |- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )
6		ismrcd.m	. . . 4 |- ( ( ph /\ x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) )
7		ismrcd.i	. . . 4 |- ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) )
8	4, 1, 5, 6, 7	ismrcd1 37261	. . 3 |- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )
9		eqid 2622	. . . 4 |- (mrCls ` dom ( F i^i _I ) ) = (mrCls ` dom ( F i^i _I ) )
10	9	mrcf 16269	. . 3 |- ( dom ( F i^i _I ) e. (Moore ` B ) -> (mrCls ` dom ( F i^i _I ) ) : ~P B --> dom ( F i^i _I ) )
11		ffn 6045	. . 3 |- ( (mrCls ` dom ( F i^i _I ) ) : ~P B --> dom ( F i^i _I ) -> (mrCls ` dom ( F i^i _I ) ) Fn ~P B )
12	8, 10, 11	3syl 18	. 2 |- ( ph -> (mrCls ` dom ( F i^i _I ) ) Fn ~P B )
13	8, 9	mrcssvd 16283	. . . . . 6 |- ( ph -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B )
14	13	adantr 481	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B )
15		elpwi 4168	. . . . . 6 |- ( z e. ~P B -> z C_ B )
16	9	mrcssid 16277	. . . . . 6 |- ( ( dom ( F i^i _I ) e. (Moore ` B ) /\ z C_ B ) -> z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) )
17	8, 15, 16	syl2an 494	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) )
18	6	3expib 1268	. . . . . . . 8 |- ( ph -> ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
19	18	alrimivv 1856	. . . . . . 7 |- ( ph -> A. y A. x ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) )
20		vex 3203	. . . . . . . 8 |- z e. _V
21		fvex 6201	. . . . . . . 8 |- ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. _V
22		sseq1 3626	. . . . . . . . . . . 12 |- ( x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) -> ( x C_ B <-> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B ) )
23	22	adantl 482	. . . . . . . . . . 11 |- ( ( y = z /\ x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( x C_ B <-> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B ) )
24		sseq12 3628	. . . . . . . . . . 11 |- ( ( y = z /\ x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( y C_ x <-> z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) )
25	23, 24	anbi12d 747	. . . . . . . . . 10 |- ( ( y = z /\ x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( ( x C_ B /\ y C_ x ) <-> ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
26		fveq2 6191	. . . . . . . . . . 11 |- ( y = z -> ( F ` y ) = ( F ` z ) )
27		fveq2 6191	. . . . . . . . . . 11 |- ( x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) -> ( F ` x ) = ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) )
28		sseq12 3628	. . . . . . . . . . 11 |- ( ( ( F ` y ) = ( F ` z ) /\ ( F ` x ) = ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) -> ( ( F ` y ) C_ ( F ` x ) <-> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
29	26, 27, 28	syl2an 494	. . . . . . . . . 10 |- ( ( y = z /\ x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( ( F ` y ) C_ ( F ` x ) <-> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
30	25, 29	imbi12d 334	. . . . . . . . 9 |- ( ( y = z /\ x = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) <-> ( ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) ) )
31	30	spc2gv 3296	. . . . . . . 8 |- ( ( z e. _V /\ ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. _V ) -> ( A. y A. x ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) -> ( ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) ) )
32	20, 21, 31	mp2an 708	. . . . . . 7 |- ( A. y A. x ( ( x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) ) -> ( ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
33	19, 32	syl 17	. . . . . 6 |- ( ph -> ( ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
34	33	adantr 481	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B /\ z C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) ) )
35	14, 17, 34	mp2and 715	. . . 4 |- ( ( ph /\ z e. ~P B ) -> ( F ` z ) C_ ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) )
36	9	mrccl 16271	. . . . . 6 |- ( ( dom ( F i^i _I ) e. (Moore ` B ) /\ z C_ B ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. dom ( F i^i _I ) )
37	8, 15, 36	syl2an 494	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. dom ( F i^i _I ) )
38	3	adantr 481	. . . . . 6 |- ( ( ph /\ z e. ~P B ) -> F Fn ~P B )
39	21	elpw 4164	. . . . . . . 8 |- ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. ~P B <-> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ B )
40	13, 39	sylibr 224	. . . . . . 7 |- ( ph -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. ~P B )
41	40	adantr 481	. . . . . 6 |- ( ( ph /\ z e. ~P B ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. ~P B )
42		fnelfp 6441	. . . . . 6 |- ( ( F Fn ~P B /\ ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. ~P B ) -> ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. dom ( F i^i _I ) <-> ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) )
43	38, 41, 42	syl2anc 693	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( ( (mrCls ` dom ( F i^i _I ) ) ` z ) e. dom ( F i^i _I ) <-> ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) = ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) )
44	37, 43	mpbid 222	. . . 4 |- ( ( ph /\ z e. ~P B ) -> ( F ` ( (mrCls ` dom ( F i^i _I ) ) ` z ) ) = ( (mrCls ` dom ( F i^i _I ) ) ` z ) )
45	35, 44	sseqtrd 3641	. . 3 |- ( ( ph /\ z e. ~P B ) -> ( F ` z ) C_ ( (mrCls ` dom ( F i^i _I ) ) ` z ) )
46	8	adantr 481	. . . 4 |- ( ( ph /\ z e. ~P B ) -> dom ( F i^i _I ) e. (Moore ` B ) )
47		sseq1 3626	. . . . . . . 8 |- ( x = z -> ( x C_ B <-> z C_ B ) )
48	47	anbi2d 740	. . . . . . 7 |- ( x = z -> ( ( ph /\ x C_ B ) <-> ( ph /\ z C_ B ) ) )
49		id 22	. . . . . . . 8 |- ( x = z -> x = z )
50		fveq2 6191	. . . . . . . 8 |- ( x = z -> ( F ` x ) = ( F ` z ) )
51	49, 50	sseq12d 3634	. . . . . . 7 |- ( x = z -> ( x C_ ( F ` x ) <-> z C_ ( F ` z ) ) )
52	48, 51	imbi12d 334	. . . . . 6 |- ( x = z -> ( ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) ) <-> ( ( ph /\ z C_ B ) -> z C_ ( F ` z ) ) ) )
53	52, 5	chvarv 2263	. . . . 5 |- ( ( ph /\ z C_ B ) -> z C_ ( F ` z ) )
54	15, 53	sylan2 491	. . . 4 |- ( ( ph /\ z e. ~P B ) -> z C_ ( F ` z ) )
55	50	fveq2d 6195	. . . . . . . . 9 |- ( x = z -> ( F ` ( F ` x ) ) = ( F ` ( F ` z ) ) )
56	55, 50	eqeq12d 2637	. . . . . . . 8 |- ( x = z -> ( ( F ` ( F ` x ) ) = ( F ` x ) <-> ( F ` ( F ` z ) ) = ( F ` z ) ) )
57	48, 56	imbi12d 334	. . . . . . 7 |- ( x = z -> ( ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) ) <-> ( ( ph /\ z C_ B ) -> ( F ` ( F ` z ) ) = ( F ` z ) ) ) )
58	57, 7	chvarv 2263	. . . . . 6 |- ( ( ph /\ z C_ B ) -> ( F ` ( F ` z ) ) = ( F ` z ) )
59	15, 58	sylan2 491	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( F ` ( F ` z ) ) = ( F ` z ) )
60	1	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ z e. ~P B ) -> ( F ` z ) e. ~P B )
61		fnelfp 6441	. . . . . 6 |- ( ( F Fn ~P B /\ ( F ` z ) e. ~P B ) -> ( ( F ` z ) e. dom ( F i^i _I ) <-> ( F ` ( F ` z ) ) = ( F ` z ) ) )
62	38, 60, 61	syl2anc 693	. . . . 5 |- ( ( ph /\ z e. ~P B ) -> ( ( F ` z ) e. dom ( F i^i _I ) <-> ( F ` ( F ` z ) ) = ( F ` z ) ) )
63	59, 62	mpbird 247	. . . 4 |- ( ( ph /\ z e. ~P B ) -> ( F ` z ) e. dom ( F i^i _I ) )
64	9	mrcsscl 16280	. . . 4 |- ( ( dom ( F i^i _I ) e. (Moore ` B ) /\ z C_ ( F ` z ) /\ ( F ` z ) e. dom ( F i^i _I ) ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ ( F ` z ) )
65	46, 54, 63, 64	syl3anc 1326	. . 3 |- ( ( ph /\ z e. ~P B ) -> ( (mrCls ` dom ( F i^i _I ) ) ` z ) C_ ( F ` z ) )
66	45, 65	eqssd 3620	. 2 |- ( ( ph /\ z e. ~P B ) -> ( F ` z ) = ( (mrCls ` dom ( F i^i _I ) ) ` z ) )
67	3, 12, 66	eqfnfvd 6314	1 |- ( ph -> F = (mrCls ` dom ( F i^i _I ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   _I cid 5023   dom cdm 5114   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:  istopclsd  37263  ismrc  37264