Theorem kqcldsat 20803

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 20787). (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 20795	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 6030	. . . . . 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 471	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) <-> ( z e. X /\ ( F ` z ) e. ( F " U ) ) ) )
6		noel 3747	. . . . . . . 8 |- -. ( F ` z ) e. (/)
7		elin 3629	. . . . . . . . 9 |- ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
8		incom 3637	. . . . . . . . . . 11 |- ( ( F " U ) i^i ( F " ( X \ U ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) )
9		eqid 2462	. . . . . . . . . . . . . . . . . . . 20 |- U. J = U. J
10	9	cldss 20099	. . . . . . . . . . . . . . . . . . 19 |- ( U e. ( Clsd ` J ) -> U C_ U. J )
11	10	adantl 472	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ U. J )
12		fndm 5701	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn X -> dom F = X )
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> dom F = X )
14		toponuni 19997	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	eqtrd 2496	. . . . . . . . . . . . . . . . . . 19 |- ( J e. (TopOn ` X ) -> dom F = U. J )
16	15	adantr 471	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = U. J )
17	11, 16	sseqtr4d 3481	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ dom F )
18	13	adantr 471	. . . . . . . . . . . . . . . . 17 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> dom F = X )
19	17, 18	sseqtrd 3480	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ X )
20	19	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> U C_ X )
21		dfss4 3689	. . . . . . . . . . . . . . 15 |- ( U C_ X <-> ( X \ ( X \ U ) ) = U )
22	20, 21	sylib 201	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ ( X \ U ) ) = U )
23	22	imaeq2d 5190	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( F " ( X \ ( X \ U ) ) ) = ( F " U ) )
24	23	ineq2d 3646	. . . . . . . . . . . 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 765	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> J e. (TopOn ` X ) )
26	14	adantr 471	. . . . . . . . . . . . . . . 16 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> X = U. J )
27	26	difeq1d 3562	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) = ( U. J \ U ) )
28	9	cldopn 20101	. . . . . . . . . . . . . . . 16 |- ( U e. ( Clsd ` J ) -> ( U. J \ U ) e. J )
29	28	adantl 472	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( U. J \ U ) e. J )
30	27, 29	eqeltrd 2540	. . . . . . . . . . . . . 14 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) e. J )
31	30	adantr 471	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( X \ U ) e. J )
32	1	kqdisj 20802	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ ( X \ U ) e. J ) -> ( ( F " ( X \ U ) ) i^i ( F " ( X \ ( X \ U ) ) ) ) = (/) )
33	25, 31, 32	syl2anc 671	. . . . . . . . . . . 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 2498	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " U ) ) = (/) )
35	8, 34	syl5eq 2508	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
36	35	eleq2d 2525	. . . . . . . . 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 267	. . . . . . . 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 309	. . . . . . 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 428	. . . . . . 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 217	. . . . . 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 3426	. . . . . . . . . 10 |- ( z e. ( X \ U ) <-> ( z e. X /\ -. z e. U ) )
42	41	baibr 920	. . . . . . . . 9 |- ( z e. X -> ( -. z e. U <-> z e. ( X \ U ) ) )
43	42	adantl 472	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> z e. ( X \ U ) ) )
44		simpr 467	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> z e. X )
45	1	kqfvima 20800	. . . . . . . . 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 1276	. . . . . . . 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 261	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( -. z e. U <-> ( F ` z ) e. ( F " ( X \ U ) ) ) )
48	47	con1bid 336	. . . . . 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 222	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F ` z ) e. ( F " U ) -> z e. U ) )
50	49	expimpd 612	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( ( z e. X /\ ( F ` z ) e. ( F " U ) ) -> z e. U ) )
51	5, 50	sylbid 223	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( z e. ( `' F " ( F " U ) ) -> z e. U ) )
52	51	ssrdv 3450	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) C_ U )
53		dfss1 3649	. . . 4 |- ( U C_ dom F <-> ( dom F i^i U ) = U )
54	17, 53	sylib 201	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( dom F i^i U ) = U )
55		dminss 5272	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
56	54, 55	syl6eqssr 3495	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ ( `' F " ( F " U ) ) )
57	52, 56	eqssd 3461	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   {crab 2753   \ cdif 3413   i^i cin 3415   C_ wss 3416   (/)c0 3743   U.cuni 4212   |-> cmpt 4477   `'ccnv 4855   dom cdm 4856   "cima 4859   Fn wfn 5600   ` cfv 5605  TopOnctopon 19973   Clsdccld 20086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fv 5613  df-top 19976  df-topon 19978  df-cld 20089

This theorem is referenced by:  kqcld  20805