Theorem ntrclsk2 38366

Description: An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
ntrcls.d	|- D = ( O ` B )
ntrcls.r	|- ( ph -> I D K )

Assertion

Ref	Expression
ntrclsk2	|- ( ph -> ( A. s e. ~P B ( I ` s ) C_ s <-> A. s e. ~P B s C_ ( K ` s ) ) )

Distinct variable groups:   B, i, j, k, s   j, I, k, s   ph, i, j, k, s

Allowed substitution hints:   D( i, j, k, s)   I( i)   K( i, j, k, s)   O( i, j, k, s)

Proof of Theorem ntrclsk2

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( s = t -> ( I ` s ) = ( I ` t ) )
2		id 22	. . . 4 |- ( s = t -> s = t )
3	1, 2	sseq12d 3634	. . 3 |- ( s = t -> ( ( I ` s ) C_ s <-> ( I ` t ) C_ t ) )
4	3	cbvralv 3171	. 2 |- ( A. s e. ~P B ( I ` s ) C_ s <-> A. t e. ~P B ( I ` t ) C_ t )
5		ntrcls.d	. . . . 5 |- D = ( O ` B )
6		ntrcls.r	. . . . 5 |- ( ph -> I D K )
7	5, 6	ntrclsrcomplex 38333	. . . 4 |- ( ph -> ( B \ s ) e. ~P B )
8	7	adantr 481	. . 3 |- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B )
9	5, 6	ntrclsrcomplex 38333	. . . . 5 |- ( ph -> ( B \ t ) e. ~P B )
10	9	adantr 481	. . . 4 |- ( ( ph /\ t e. ~P B ) -> ( B \ t ) e. ~P B )
11		difeq2 3722	. . . . . 6 |- ( s = ( B \ t ) -> ( B \ s ) = ( B \ ( B \ t ) ) )
12	11	eqeq2d 2632	. . . . 5 |- ( s = ( B \ t ) -> ( t = ( B \ s ) <-> t = ( B \ ( B \ t ) ) ) )
13	12	adantl 482	. . . 4 |- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> ( t = ( B \ s ) <-> t = ( B \ ( B \ t ) ) ) )
14		elpwi 4168	. . . . . . 7 |- ( t e. ~P B -> t C_ B )
15		dfss4 3858	. . . . . . 7 |- ( t C_ B <-> ( B \ ( B \ t ) ) = t )
16	14, 15	sylib 208	. . . . . 6 |- ( t e. ~P B -> ( B \ ( B \ t ) ) = t )
17	16	adantl 482	. . . . 5 |- ( ( ph /\ t e. ~P B ) -> ( B \ ( B \ t ) ) = t )
18	17	eqcomd 2628	. . . 4 |- ( ( ph /\ t e. ~P B ) -> t = ( B \ ( B \ t ) ) )
19	10, 13, 18	rspcedvd 3317	. . 3 |- ( ( ph /\ t e. ~P B ) -> E. s e. ~P B t = ( B \ s ) )
20		fveq2 6191	. . . . . 6 |- ( t = ( B \ s ) -> ( I ` t ) = ( I ` ( B \ s ) ) )
21		id 22	. . . . . 6 |- ( t = ( B \ s ) -> t = ( B \ s ) )
22	20, 21	sseq12d 3634	. . . . 5 |- ( t = ( B \ s ) -> ( ( I ` t ) C_ t <-> ( I ` ( B \ s ) ) C_ ( B \ s ) ) )
23	22	3ad2ant3 1084	. . . 4 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( I ` t ) C_ t <-> ( I ` ( B \ s ) ) C_ ( B \ s ) ) )
24		ntrcls.o	. . . . . . . . . . . 12 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
25	24, 5, 6	ntrclsiex 38351	. . . . . . . . . . 11 |- ( ph -> I e. ( ~P B ^m ~P B ) )
26		elmapi 7879	. . . . . . . . . . 11 |- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B )
27	25, 26	syl 17	. . . . . . . . . 10 |- ( ph -> I : ~P B --> ~P B )
28	27	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> I : ~P B --> ~P B )
29	7	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( B \ s ) e. ~P B )
30	28, 29	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( I ` ( B \ s ) ) e. ~P B )
31	30	elpwid 4170	. . . . . . 7 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( I ` ( B \ s ) ) C_ B )
32		difssd 3738	. . . . . . 7 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( B \ s ) C_ B )
33		sscon34b 38317	. . . . . . 7 |- ( ( ( I ` ( B \ s ) ) C_ B /\ ( B \ s ) C_ B ) -> ( ( I ` ( B \ s ) ) C_ ( B \ s ) <-> ( B \ ( B \ s ) ) C_ ( B \ ( I ` ( B \ s ) ) ) ) )
34	31, 32, 33	syl2anc 693	. . . . . 6 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( I ` ( B \ s ) ) C_ ( B \ s ) <-> ( B \ ( B \ s ) ) C_ ( B \ ( I ` ( B \ s ) ) ) ) )
35		simp2 1062	. . . . . . 7 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> s e. ~P B )
36		elpwi 4168	. . . . . . . . 9 |- ( s e. ~P B -> s C_ B )
37		dfss4 3858	. . . . . . . . 9 |- ( s C_ B <-> ( B \ ( B \ s ) ) = s )
38	36, 37	sylib 208	. . . . . . . 8 |- ( s e. ~P B -> ( B \ ( B \ s ) ) = s )
39	38	sseq1d 3632	. . . . . . 7 |- ( s e. ~P B -> ( ( B \ ( B \ s ) ) C_ ( B \ ( I ` ( B \ s ) ) ) <-> s C_ ( B \ ( I ` ( B \ s ) ) ) ) )
40	35, 39	syl 17	. . . . . 6 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( B \ ( B \ s ) ) C_ ( B \ ( I ` ( B \ s ) ) ) <-> s C_ ( B \ ( I ` ( B \ s ) ) ) ) )
41	34, 40	bitrd 268	. . . . 5 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( I ` ( B \ s ) ) C_ ( B \ s ) <-> s C_ ( B \ ( I ` ( B \ s ) ) ) ) )
42	5, 6	ntrclsbex 38332	. . . . . . . 8 |- ( ph -> B e. _V )
43	42	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> B e. _V )
44	25	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> I e. ( ~P B ^m ~P B ) )
45		eqid 2622	. . . . . . 7 |- ( D ` I ) = ( D ` I )
46		eqid 2622	. . . . . . 7 |- ( ( D ` I ) ` s ) = ( ( D ` I ) ` s )
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 38313	. . . . . 6 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( D ` I ) ` s ) = ( B \ ( I ` ( B \ s ) ) ) )
48	47	sseq2d 3633	. . . . 5 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( s C_ ( ( D ` I ) ` s ) <-> s C_ ( B \ ( I ` ( B \ s ) ) ) ) )
49	24, 5, 6	ntrclsfv1 38353	. . . . . . . 8 |- ( ph -> ( D ` I ) = K )
50	49	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( D ` I ) ` s ) = ( K ` s ) )
51	50	sseq2d 3633	. . . . . 6 |- ( ph -> ( s C_ ( ( D ` I ) ` s ) <-> s C_ ( K ` s ) ) )
52	51	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( s C_ ( ( D ` I ) ` s ) <-> s C_ ( K ` s ) ) )
53	41, 48, 52	3bitr2d 296	. . . 4 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( I ` ( B \ s ) ) C_ ( B \ s ) <-> s C_ ( K ` s ) ) )
54	23, 53	bitrd 268	. . 3 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( ( I ` t ) C_ t <-> s C_ ( K ` s ) ) )
55	8, 19, 54	ralxfrd2 4884	. 2 |- ( ph -> ( A. t e. ~P B ( I ` t ) C_ t <-> A. s e. ~P B s C_ ( K ` s ) ) )
56	4, 55	syl5bb 272	1 |- ( ph -> ( A. s e. ~P B ( I ` s ) C_ s <-> A. s e. ~P B s C_ ( K ` s ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-rep 4771  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-reu 2919  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)