Theorem r0cld 21541

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from A is closed. (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
r0cld	|- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> { z e. X | A. o e. J ( z e. o <-> A e. o ) } e. ( Clsd ` J ) )

Distinct variable groups:   x, o, y, z, A   o, J, x, y, z   o, F, z   o, X, x, y, z

Allowed substitution hints:   F( x, y)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqffn 21528	. . . . 5 |- ( J e. (TopOn ` X ) -> F Fn X )
3	2	3ad2ant1 1082	. . . 4 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> F Fn X )
4		fncnvima2 6339	. . . 4 |- ( F Fn X -> ( `' F " { ( F ` A ) } ) = { z e. X | ( F ` z ) e. { ( F ` A ) } } )
5	3, 4	syl 17	. . 3 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ( `' F " { ( F ` A ) } ) = { z e. X | ( F ` z ) e. { ( F ` A ) } } )
6		fvex 6201	. . . . . 6 |- ( F ` z ) e. _V
7	6	elsn 4192	. . . . 5 |- ( ( F ` z ) e. { ( F ` A ) } <-> ( F ` z ) = ( F ` A ) )
8		simpl1 1064	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) /\ z e. X ) -> J e. (TopOn ` X ) )
9		simpr 477	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) /\ z e. X ) -> z e. X )
10		simpl3 1066	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) /\ z e. X ) -> A e. X )
11	1	kqfeq 21527	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ z e. X /\ A e. X ) -> ( ( F ` z ) = ( F ` A ) <-> A. y e. J ( z e. y <-> A e. y ) ) )
12		eleq2 2690	. . . . . . . . 9 |- ( y = o -> ( z e. y <-> z e. o ) )
13		eleq2 2690	. . . . . . . . 9 |- ( y = o -> ( A e. y <-> A e. o ) )
14	12, 13	bibi12d 335	. . . . . . . 8 |- ( y = o -> ( ( z e. y <-> A e. y ) <-> ( z e. o <-> A e. o ) ) )
15	14	cbvralv 3171	. . . . . . 7 |- ( A. y e. J ( z e. y <-> A e. y ) <-> A. o e. J ( z e. o <-> A e. o ) )
16	11, 15	syl6bb 276	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ z e. X /\ A e. X ) -> ( ( F ` z ) = ( F ` A ) <-> A. o e. J ( z e. o <-> A e. o ) ) )
17	8, 9, 10, 16	syl3anc 1326	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) /\ z e. X ) -> ( ( F ` z ) = ( F ` A ) <-> A. o e. J ( z e. o <-> A e. o ) ) )
18	7, 17	syl5bb 272	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) /\ z e. X ) -> ( ( F ` z ) e. { ( F ` A ) } <-> A. o e. J ( z e. o <-> A e. o ) ) )
19	18	rabbidva 3188	. . 3 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> { z e. X | ( F ` z ) e. { ( F ` A ) } } = { z e. X | A. o e. J ( z e. o <-> A e. o ) } )
20	5, 19	eqtrd 2656	. 2 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ( `' F " { ( F ` A ) } ) = { z e. X | A. o e. J ( z e. o <-> A e. o ) } )
21	1	kqid 21531	. . . 4 |- ( J e. (TopOn ` X ) -> F e. ( J Cn (KQ ` J ) ) )
22	21	3ad2ant1 1082	. . 3 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> F e. ( J Cn (KQ ` J ) ) )
23		simp2 1062	. . . 4 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> (KQ ` J ) e. Fre )
24		simp3 1063	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> A e. X )
25		fnfvelrn 6356	. . . . . 6 |- ( ( F Fn X /\ A e. X ) -> ( F ` A ) e. ran F )
26	3, 24, 25	syl2anc 693	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ( F ` A ) e. ran F )
27	1	kqtopon 21530	. . . . . . 7 |- ( J e. (TopOn ` X ) -> (KQ ` J ) e. (TopOn ` ran F ) )
28	27	3ad2ant1 1082	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> (KQ ` J ) e. (TopOn ` ran F ) )
29		toponuni 20719	. . . . . 6 |- ( (KQ ` J ) e. (TopOn ` ran F ) -> ran F = U. (KQ ` J ) )
30	28, 29	syl 17	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ran F = U. (KQ ` J ) )
31	26, 30	eleqtrd 2703	. . . 4 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ( F ` A ) e. U. (KQ ` J ) )
32		eqid 2622	. . . . 5 |- U. (KQ ` J ) = U. (KQ ` J )
33	32	t1sncld 21130	. . . 4 |- ( ( (KQ ` J ) e. Fre /\ ( F ` A ) e. U. (KQ ` J ) ) -> { ( F ` A ) } e. ( Clsd ` (KQ ` J ) ) )
34	23, 31, 33	syl2anc 693	. . 3 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> { ( F ` A ) } e. ( Clsd ` (KQ ` J ) ) )
35		cnclima 21072	. . 3 |- ( ( F e. ( J Cn (KQ ` J ) ) /\ { ( F ` A ) } e. ( Clsd ` (KQ ` J ) ) ) -> ( `' F " { ( F ` A ) } ) e. ( Clsd ` J ) )
36	22, 34, 35	syl2anc 693	. 2 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> ( `' F " { ( F ` A ) } ) e. ( Clsd ` J ) )
37	20, 36	eqeltrrd 2702	1 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Fre /\ A e. X ) -> { z e. X | A. o e. J ( z e. o <-> A e. o ) } e. ( Clsd ` J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   Clsdccld 20820   Cn ccn 21028   Frect1 21111  KQckq 21496

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cld 20823  df-cn 21031  df-t1 21118  df-kq 21497

This theorem is referenced by:  nrmr0reg  21552