Theorem kqcldsat 19962

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 19946). (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 19954	. . . . . 6 |- ( J e. (TopOn ` X ) -> F Fn X )
3		elpreima 5992	. . . . . 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 3782	. . . . . . . 8 |- -. ( F ` z ) e. (/)
7		elin 3680	. . . . . . . . 9 |- ( ( F ` z ) e. ( ( F " U ) i^i ( F " ( X \ U ) ) ) <-> ( ( F ` z ) e. ( F " U ) /\ ( F ` z ) e. ( F " ( X \ U ) ) ) )
8		incom 3684	. . . . . . . . . . 11 |- ( ( F " U ) i^i ( F " ( X \ U ) ) ) = ( ( F " ( X \ U ) ) i^i ( F " U ) )
9		eqid 2460	. . . . . . . . . . . . . . . . . . . 20 |- U. J = U. J
10	9	cldss 19289	. . . . . . . . . . . . . . . . . . 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 5671	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn X -> dom F = X )
13	2, 12	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> dom F = X )
14		toponuni 19188	. . . . . . . . . . . . . . . . . . . 20 |- ( J e. (TopOn ` X ) -> X = U. J )
15	13, 14	eqtrd 2501	. . . . . . . . . . . . . . . . . . 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 3534	. . . . . . . . . . . . . . . . 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 3533	. . . . . . . . . . . . . . . 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 3725	. . . . . . . . . . . . . . 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 5328	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( F " ( X \ ( X \ U ) ) ) = ( F " U ) )
24	23	ineq2d 3693	. . . . . . . . . . . 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 3614	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( X \ U ) = ( U. J \ U ) )
28	9	cldopn 19291	. . . . . . . . . . . . . . . 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 2548	. . . . . . . . . . . . . 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 19961	. . . . . . . . . . . . 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 2503	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " ( X \ U ) ) i^i ( F " U ) ) = (/) )
35	8, 34	syl5eq 2513	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) /\ z e. X ) -> ( ( F " U ) i^i ( F " ( X \ U ) ) ) = (/) )
36	35	eleq2d 2530	. . . . . . . . 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 3479	. . . . . . . . . 10 |- ( z e. ( X \ U ) <-> ( z e. X /\ -. z e. U ) )
42	41	baibr 899	. . . . . . . . 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 19959	. . . . . . . . 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 1223	. . . . . . . 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 3503	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) C_ U )
53		dfss1 3696	. . . 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 5411	. . 3 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
56	54, 55	syl6eqssr 3548	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> U C_ ( `' F " ( F " U ) ) )
57	52, 56	eqssd 3514	1 |- ( ( J e. (TopOn ` X ) /\ U e. ( Clsd ` J ) ) -> ( `' F " ( F " U ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   {crab 2811   \ cdif 3466   i^i cin 3468   C_ wss 3469   (/)c0 3778   U.cuni 4238   |-> cmpt 4498   `'ccnv 4991   dom cdm 4992   "cima 4995   Fn wfn 5574   ` cfv 5579  TopOnctopon 19155   Clsdccld 19276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-top 19159  df-topon 19162  df-cld 19279

This theorem is referenced by:  kqcld  19964