Theorem kqcldsat 19439

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 19423). (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 19431	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 5933	. . . . . 6 |- ( F Fn X -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
4	2, 3	syl 16	. . . . 5 |- ( J e. (TopOn ` X ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
5	4	adantr 465	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
6		noel 3750	. . . . . . . 8 |- -. ( F ` z ) e. (/)
7		elin 3648	. . . . . . . . 9 |- ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
8		incom 3652	. . . . . . . . . . 11 |- ( ( F " U ) i^i ( F " ( X \ U ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) )
9		eqid 2454	. . . . . . . . . . . . . . . . . . . 20 |- U. J = U. J
10	9	cldss 18766	. . . . . . . . . . . . . . . . . . 19 |- ( U e. ( Clsd ` J ) -> U C_ U. J )
11	10	adantl 466	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ U. J )
12		fndm 5619	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn X -> dom F = X )
13	2, 12	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> dom F = X )
14		toponuni 18665	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	eqtrd 2495	. . . . . . . . . . . . . . . . . . 19 |- ( J e. (TopOn ` X ) -> dom F = U. J )
16	15	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = U. J )
17	11, 16	sseqtr4d 3502	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ dom F )
18	13	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = X )
19	17, 18	sseqtrd 3501	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ X )
20	19	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> U C_ X )
21		dfss4 3693	. . . . . . . . . . . . . . 15 |- ( U C_ X <-> ( X \ ( X \ U ) ) = U )
22	20, 21	sylib 196	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ ( X \ U ) ) = U )
23	22	imaeq2d 5278	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( F " ( X \ ( X \ U ) ) ) = ( F " U ) )
24	23	ineq2d 3661	. . . . . . . . . . . 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 753	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> J e. (TopOn ` X ) )
26	14	adantr 465	. . . . . . . . . . . . . . . 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 18768	. . . . . . . . . . . . . . . 16 |- ( U e. ( Clsd ` J ) -> ( U. J \ U ) e. J )
29	28	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( U. J \ U ) e. J )
30	27, 29	eqeltrd 2542	. . . . . . . . . . . . . 14 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) e. J )
31	30	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ U ) e. J )
32	1	kqdisj 19438	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ ( X \ U ) e. J ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = (/) )
33	25, 31, 32	syl2anc 661	. . . . . . . . . . . 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 2497	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " U ) ) = (/) )
35	8, 34	syl5eq 2507	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
36	35	eleq2d 2524	. . . . . . . . 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 259	. . . . . . . 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 303	. . . . . . 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 422	. . . . . . 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 212	. . . . . 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 3447	. . . . . . . . . 10 |- ( z e. ( X \ U ) <-> ( z e. X /\ -. z e. U ) )
42	41	baibr 897	. . . . . . . . 9 |- ( z e. X -> ( -. z e. U <-> z e. ( X \ U ) ) )
43	42	adantl 466	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> z e. ( X \ U ) ) )
44		simpr 461	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> z e. X )
45	1	kqfvima 19436	. . . . . . . . 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 1219	. . . . . . . 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 253	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
48	47	con1bid 330	. . . . . 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 214	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> z e. U ) )
50	49	expimpd 603	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( ( z e. X /\ ( F ` z ) e. ( F " U ) ) -> z e. U ) )
51	5, 50	sylbid 215	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) -> z e. U ) )
52	51	ssrdv 3471	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) C_ U )
53		dfss1 3664	. . . 4 |- ( U C_ dom F <-> ( dom F i^i U ) = U )
54	17, 53	sylib 196	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( dom F i^i U ) = U )
55		dminss 5360	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
56	54, 55	syl6eqssr 3516	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ ( `' F " ( F " U ) ) )
57	52, 56	eqssd 3482	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2803   \ cdif 3434   i^i cin 3436   C_ wss 3437   (/)c0 3746   U.cuni 4200   |-> cmpt 4459   `'ccnv 4948   dom cdm 4949   "cima 4952   Fn wfn 5522   ` cfv 5527  TopOnctopon 18632   Clsdccld 18753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-top 18636  df-topon 18639  df-cld 18756

This theorem is referenced by:  kqcld  19441