Theorem kqcldsat 20746

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 20730). (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	|- F = ( x e. X |-> { y e. J | x e. y } )

Assertion

Ref	Expression
kqcldsat	|- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Distinct variable groups:   x, y, J   x, X, y

Allowed substitution hints:   U( x, y)   F( x, y)

Proof of Theorem kqcldsat

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . 7 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqffn 20738	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 6017	. . . . . 6 |- ( F Fn X -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
4	2, 3	syl 17	. . . . 5 |- ( J e. (TopOn ` X ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
5	4	adantr 466	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
6		noel 3765	. . . . . . . 8 |- -. ( F ` z ) e. (/)
7		elin 3649	. . . . . . . . 9 |- ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
8		incom 3655	. . . . . . . . . . 11 |- ( ( F " U ) i^i ( F " ( X \ U ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) )
9		eqid 2422	. . . . . . . . . . . . . . . . . . . 20 |- U. J = U. J
10	9	cldss 20042	. . . . . . . . . . . . . . . . . . 19 |- ( U e. ( Clsd ` J ) -> U C_ U. J )
11	10	adantl 467	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ U. J )
12		fndm 5693	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn X -> dom F = X )
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> dom F = X )
14		toponuni 19940	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	eqtrd 2463	. . . . . . . . . . . . . . . . . . 19 |- ( J e. (TopOn ` X ) -> dom F = U. J )
16	15	adantr 466	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = U. J )
17	11, 16	sseqtr4d 3501	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ dom F )
18	13	adantr 466	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = X )
19	17, 18	sseqtrd 3500	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ X )
20	19	adantr 466	. . . . . . . . . . . . . . 15 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> U C_ X )
21		dfss4 3707	. . . . . . . . . . . . . . 15 |- ( U C_ X <-> ( X \ ( X \ U ) ) = U )
22	20, 21	sylib 199	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ ( X \ U ) ) = U )
23	22	imaeq2d 5187	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( F " ( X \ ( X \ U ) ) ) = ( F " U ) )
24	23	ineq2d 3664	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) ) )
25		simpll 758	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> J e. (TopOn ` X ) )
26	14	adantr 466	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> X = U. J )
27	26	difeq1d 3582	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) = ( U. J \ U ) )
28	9	cldopn 20044	. . . . . . . . . . . . . . . 16 |- ( U e. ( Clsd ` J ) -> ( U. J \ U ) e. J )
29	28	adantl 467	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( U. J \ U ) e. J )
30	27, 29	eqeltrd 2507	. . . . . . . . . . . . . 14 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) e. J )
31	30	adantr 466	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ U ) e. J )
32	1	kqdisj 20745	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ ( X \ U ) e. J ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = (/) )
33	25, 31, 32	syl2anc 665	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = (/) )
34	24, 33	eqtr3d 2465	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " U ) ) = (/) )
35	8, 34	syl5eq 2475	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
36	35	eleq2d 2492	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( F ` z ) e. (/) ) )
37	7, 36	syl5bbr 262	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) <-> ( F ` z ) e. (/) ) )
38	6, 37	mtbiri 304	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> -. ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
39		imnan 423	. . . . . . 7 |- ( ( ( F ` z ) e. ( F " U ) -> -. ( F ` z ) e. ( F " ( X \ U ) ) ) <-> -. ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
40	38, 39	sylibr 215	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> -. ( F ` z ) e. ( F " ( X \ U ) ) ) )
41		eldif 3446	. . . . . . . . . 10 |- ( z e. ( X \ U ) <-> ( z e. X /\ -. z e. U ) )
42	41	baibr 912	. . . . . . . . 9 |- ( z e. X -> ( -. z e. U <-> z e. ( X \ U ) ) )
43	42	adantl 467	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> z e. ( X \ U ) ) )
44		simpr 462	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> z e. X )
45	1	kqfvima 20743	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ ( X \ U ) e. J /\ z e. X ) -> ( z e. ( X \ U ) <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
46	25, 31, 44, 45	syl3anc 1264	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( z e. ( X \ U ) <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
47	43, 46	bitrd 256	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
48	47	con1bid 331	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. ( F ` z ) e. ( F " ( X \ U ) ) <-> z e. U ) )
49	40, 48	sylibd 217	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> z e. U ) )
50	49	expimpd 606	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( ( z e. X /\ ( F ` z ) e. ( F " U ) ) -> z e. U ) )
51	5, 50	sylbid 218	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) -> z e. U ) )
52	51	ssrdv 3470	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) C_ U )
53		dfss1 3667	. . . 4 |- ( U C_ dom F <-> ( dom F i^i U ) = U )
54	17, 53	sylib 199	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( dom F i^i U ) = U )
55		dminss 5269	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
56	54, 55	syl6eqssr 3515	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ ( `' F " ( F " U ) ) )
57	52, 56	eqssd 3481	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   {crab 2775   \ cdif 3433   i^i cin 3435   C_ wss 3436   (/)c0 3761   U.cuni 4219   |-> cmpt 4482   `'ccnv 4852   dom cdm 4853   "cima 4856   Fn wfn 5596   ` cfv 5601  TopOnctopon 19916   Clsdccld 20029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-top 19919  df-topon 19921  df-cld 20032

This theorem is referenced by:  kqcld  20748