Theorem kqcldsat 21536

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21520). (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 21528	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 6337	. . . . . 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 481	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
6		noel 3919	. . . . . . . 8 |- -. ( F ` z ) e. (/)
7		elin 3796	. . . . . . . . 9 |- ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
8		incom 3805	. . . . . . . . . . 11 |- ( ( F " U ) i^i ( F " ( X \ U ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) )
9		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 |- U. J = U. J
10	9	cldss 20833	. . . . . . . . . . . . . . . . . . 19 |- ( U e. ( Clsd ` J ) -> U C_ U. J )
11	10	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ U. J )
12		fndm 5990	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn X -> dom F = X )
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> dom F = X )
14		toponuni 20719	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( J e. (TopOn ` X ) -> dom F = U. J )
16	15	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = U. J )
17	11, 16	sseqtr4d 3642	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ dom F )
18	13	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = X )
19	17, 18	sseqtrd 3641	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ X )
20	19	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> U C_ X )
21		dfss4 3858	. . . . . . . . . . . . . . 15 |- ( U C_ X <-> ( X \ ( X \ U ) ) = U )
22	20, 21	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ ( X \ U ) ) = U )
23	22	imaeq2d 5466	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( F " ( X \ ( X \ U ) ) ) = ( F " U ) )
24	23	ineq2d 3814	. . . . . . . . . . . 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 790	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> J e. (TopOn ` X ) )
26	14	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> X = U. J )
27	26	difeq1d 3727	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) = ( U. J \ U ) )
28	9	cldopn 20835	. . . . . . . . . . . . . . . 16 |- ( U e. ( Clsd ` J ) -> ( U. J \ U ) e. J )
29	28	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( U. J \ U ) e. J )
30	27, 29	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) e. J )
31	30	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ U ) e. J )
32	1	kqdisj 21535	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ ( X \ U ) e. J ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = (/) )
33	25, 31, 32	syl2anc 693	. . . . . . . . . . . 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 2658	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " U ) ) = (/) )
35	8, 34	syl5eq 2668	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
36	35	eleq2d 2687	. . . . . . . . 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 274	. . . . . . . 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 317	. . . . . . 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 438	. . . . . . 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 224	. . . . . 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 3584	. . . . . . . . . 10 |- ( z e. ( X \ U ) <-> ( z e. X /\ -. z e. U ) )
42	41	baibr 945	. . . . . . . . 9 |- ( z e. X -> ( -. z e. U <-> z e. ( X \ U ) ) )
43	42	adantl 482	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> z e. ( X \ U ) ) )
44		simpr 477	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> z e. X )
45	1	kqfvima 21533	. . . . . . . . 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 1326	. . . . . . . 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 268	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
48	47	con1bid 345	. . . . . 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 229	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> z e. U ) )
50	49	expimpd 629	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( ( z e. X /\ ( F ` z ) e. ( F " U ) ) -> z e. U ) )
51	5, 50	sylbid 230	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) -> z e. U ) )
52	51	ssrdv 3609	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) C_ U )
53		sseqin2 3817	. . . 4 |- ( U C_ dom F <-> ( dom F i^i U ) = U )
54	17, 53	sylib 208	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( dom F i^i U ) = U )
55		dminss 5547	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
56	54, 55	syl6eqssr 3656	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ ( `' F " ( F " U ) ) )
57	52, 56	eqssd 3620	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   ` cfv 5888  TopOnctopon 20715   Clsdccld 20820

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-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-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-top 20699  df-topon 20716  df-cld 20823

This theorem is referenced by:  kqcld  21538