Theorem clsk1indlem4 38342

Description: The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )

Assertion

Ref	Expression
clsk1indlem4	|- A. s e. ~P 3o ( K ` ( K ` s ) ) = ( K ` s )

Distinct variable group:   s, r

Allowed substitution hints:   K( s, r)

Proof of Theorem clsk1indlem4

Step	Hyp	Ref	Expression
1		tpex 6957	. . . . . . . . . 10 |- { (/) , 1o , 2o } e. _V
2	1	a1i 11	. . . . . . . . 9 |- ( T. -> { (/) , 1o , 2o } e. _V )
3		snsstp1 4347	. . . . . . . . . . . 12 |- { (/) } C_ { (/) , 1o , 2o }
4	3	a1i 11	. . . . . . . . . . 11 |- ( T. -> { (/) } C_ { (/) , 1o , 2o } )
5		0ex 4790	. . . . . . . . . . . 12 |- (/) e. _V
6	5	snss 4316	. . . . . . . . . . 11 |- ( (/) e. { (/) , 1o , 2o } <-> { (/) } C_ { (/) , 1o , 2o } )
7	4, 6	sylibr 224	. . . . . . . . . 10 |- ( T. -> (/) e. { (/) , 1o , 2o } )
8		snsstp2 4348	. . . . . . . . . . . 12 |- { 1o } C_ { (/) , 1o , 2o }
9	8	a1i 11	. . . . . . . . . . 11 |- ( T. -> { 1o } C_ { (/) , 1o , 2o } )
10		1on 7567	. . . . . . . . . . . . 13 |- 1o e. On
11	10	elexi 3213	. . . . . . . . . . . 12 |- 1o e. _V
12	11	snss 4316	. . . . . . . . . . 11 |- ( 1o e. { (/) , 1o , 2o } <-> { 1o } C_ { (/) , 1o , 2o } )
13	9, 12	sylibr 224	. . . . . . . . . 10 |- ( T. -> 1o e. { (/) , 1o , 2o } )
14	7, 13	prssd 4354	. . . . . . . . 9 |- ( T. -> { (/) , 1o } C_ { (/) , 1o , 2o } )
15	2, 14	sselpwd 4807	. . . . . . . 8 |- ( T. -> { (/) , 1o } e. ~P { (/) , 1o , 2o } )
16	15	trud 1493	. . . . . . 7 |- { (/) , 1o } e. ~P { (/) , 1o , 2o }
17		df3o2 38322	. . . . . . . 8 |- 3o = { (/) , 1o , 2o }
18	17	pweqi 4162	. . . . . . 7 |- ~P 3o = ~P { (/) , 1o , 2o }
19	16, 18	eleqtrri 2700	. . . . . 6 |- { (/) , 1o } e. ~P 3o
20	19	a1i 11	. . . . 5 |- ( s e. ~P 3o -> { (/) , 1o } e. ~P 3o )
21		id 22	. . . . 5 |- ( s e. ~P 3o -> s e. ~P 3o )
22	20, 21	ifcld 4131	. . . 4 |- ( s e. ~P 3o -> if ( s = { (/) } , { (/) , 1o } , s ) e. ~P 3o )
23		eqeq1 2626	. . . . . . . 8 |- ( r = if ( s = { (/) } , { (/) , 1o } , s ) -> ( r = { (/) } <-> if ( s = { (/) } , { (/) , 1o } , s ) = { (/) } ) )
24		eqcom 2629	. . . . . . . . 9 |- ( if ( s = { (/) } , { (/) , 1o } , s ) = { (/) } <-> { (/) } = if ( s = { (/) } , { (/) , 1o } , s ) )
25		eqif 4126	. . . . . . . . 9 |- ( { (/) } = if ( s = { (/) } , { (/) , 1o } , s ) <-> ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) ) )
26	24, 25	bitri 264	. . . . . . . 8 |- ( if ( s = { (/) } , { (/) , 1o } , s ) = { (/) } <-> ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) ) )
27	23, 26	syl6bb 276	. . . . . . 7 |- ( r = if ( s = { (/) } , { (/) , 1o } , s ) -> ( r = { (/) } <-> ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) ) ) )
28		id 22	. . . . . . 7 |- ( r = if ( s = { (/) } , { (/) , 1o } , s ) -> r = if ( s = { (/) } , { (/) , 1o } , s ) )
29	27, 28	ifbieq2d 4111	. . . . . 6 |- ( r = if ( s = { (/) } , { (/) , 1o } , s ) -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) ) , { (/) , 1o } , if ( s = { (/) } , { (/) , 1o } , s ) ) )
30		1n0 7575	. . . . . . . . . 10 |- 1o =/= (/)
31		dfsn2 4190	. . . . . . . . . . . 12 |- { (/) } = { (/) , (/) }
32	31	eqeq1i 2627	. . . . . . . . . . 11 |- ( { (/) } = { (/) , 1o } <-> { (/) , (/) } = { (/) , 1o } )
33	5	a1i 11	. . . . . . . . . . . . 13 |- ( T. -> (/) e. _V )
34	10	a1i 11	. . . . . . . . . . . . 13 |- ( T. -> 1o e. On )
35	33, 34	preq2b 4378	. . . . . . . . . . . 12 |- ( T. -> ( { (/) , (/) } = { (/) , 1o } <-> (/) = 1o ) )
36	35	trud 1493	. . . . . . . . . . 11 |- ( { (/) , (/) } = { (/) , 1o } <-> (/) = 1o )
37		eqcom 2629	. . . . . . . . . . 11 |- ( (/) = 1o <-> 1o = (/) )
38	32, 36, 37	3bitri 286	. . . . . . . . . 10 |- ( { (/) } = { (/) , 1o } <-> 1o = (/) )
39	30, 38	nemtbir 2889	. . . . . . . . 9 |- -. { (/) } = { (/) , 1o }
40	39	intnan 960	. . . . . . . 8 |- -. ( s = { (/) } /\ { (/) } = { (/) , 1o } )
41		pm3.24 926	. . . . . . . . 9 |- -. ( s = { (/) } /\ -. s = { (/) } )
42		eqcom 2629	. . . . . . . . . 10 |- ( s = { (/) } <-> { (/) } = s )
43	42	anbi2ci 732	. . . . . . . . 9 |- ( ( s = { (/) } /\ -. s = { (/) } ) <-> ( -. s = { (/) } /\ { (/) } = s ) )
44	41, 43	mtbi 312	. . . . . . . 8 |- -. ( -. s = { (/) } /\ { (/) } = s )
45	40, 44	pm3.2ni 899	. . . . . . 7 |- -. ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) )
46	45	iffalsei 4096	. . . . . 6 |- if ( ( ( s = { (/) } /\ { (/) } = { (/) , 1o } ) \/ ( -. s = { (/) } /\ { (/) } = s ) ) , { (/) , 1o } , if ( s = { (/) } , { (/) , 1o } , s ) ) = if ( s = { (/) } , { (/) , 1o } , s )
47	29, 46	syl6eq 2672	. . . . 5 |- ( r = if ( s = { (/) } , { (/) , 1o } , s ) -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( s = { (/) } , { (/) , 1o } , s ) )
48		clsk1indlem.k	. . . . 5 |- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )
49		prex 4909	. . . . . 6 |- { (/) , 1o } e. _V
50		vex 3203	. . . . . 6 |- s e. _V
51	49, 50	ifex 4156	. . . . 5 |- if ( s = { (/) } , { (/) , 1o } , s ) e. _V
52	47, 48, 51	fvmpt 6282	. . . 4 |- ( if ( s = { (/) } , { (/) , 1o } , s ) e. ~P 3o -> ( K ` if ( s = { (/) } , { (/) , 1o } , s ) ) = if ( s = { (/) } , { (/) , 1o } , s ) )
53	22, 52	syl 17	. . 3 |- ( s e. ~P 3o -> ( K ` if ( s = { (/) } , { (/) , 1o } , s ) ) = if ( s = { (/) } , { (/) , 1o } , s ) )
54		eqeq1 2626	. . . . . 6 |- ( r = s -> ( r = { (/) } <-> s = { (/) } ) )
55		id 22	. . . . . 6 |- ( r = s -> r = s )
56	54, 55	ifbieq2d 4111	. . . . 5 |- ( r = s -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( s = { (/) } , { (/) , 1o } , s ) )
57	56, 48, 51	fvmpt 6282	. . . 4 |- ( s e. ~P 3o -> ( K ` s ) = if ( s = { (/) } , { (/) , 1o } , s ) )
58	57	fveq2d 6195	. . 3 |- ( s e. ~P 3o -> ( K ` ( K ` s ) ) = ( K ` if ( s = { (/) } , { (/) , 1o } , s ) ) )
59	53, 58, 57	3eqtr4d 2666	. 2 |- ( s e. ~P 3o -> ( K ` ( K ` s ) ) = ( K ` s ) )
60	59	rgen 2922	1 |- A. s e. ~P 3o ( K ` ( K ` s ) ) = ( K ` s )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   {ctp 4181   |-> cmpt 4729   Oncon0 5723   ` cfv 5888   1oc1o 7553   2oc2o 7554   3oc3o 7555

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-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by:  clsk1independent  38344