Theorem restlp 20987

Description: The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypotheses

Ref	Expression
restcls.1	|- X = U. J
restcls.2	|- K = ( J ↾t Y )

Assertion

Ref	Expression
restlp	|- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( ( limPt ` K ) ` S ) = ( ( ( limPt ` J ) ` S ) i^i Y ) )

Proof of Theorem restlp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . . . . . 7 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> S C_ Y )
2	1	ssdifssd 3748	. . . . . 6 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( S \ { x } ) C_ Y )
3		restcls.1	. . . . . . 7 |- X = U. J
4		restcls.2	. . . . . . 7 |- K = ( J ↾t Y )
5	3, 4	restcls 20985	. . . . . 6 |- ( ( J e. Top /\ Y C_ X /\ ( S \ { x } ) C_ Y ) -> ( ( cls ` K ) ` ( S \ { x } ) ) = ( ( ( cls ` J ) ` ( S \ { x } ) ) i^i Y ) )
6	2, 5	syld3an3 1371	. . . . 5 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( ( cls ` K ) ` ( S \ { x } ) ) = ( ( ( cls ` J ) ` ( S \ { x } ) ) i^i Y ) )
7	6	eleq2d 2687	. . . 4 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( cls ` K ) ` ( S \ { x } ) ) <-> x e. ( ( ( cls ` J ) ` ( S \ { x } ) ) i^i Y ) ) )
8		elin 3796	. . . 4 |- ( x e. ( ( ( cls ` J ) ` ( S \ { x } ) ) i^i Y ) <-> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) /\ x e. Y ) )
9	7, 8	syl6bb 276	. . 3 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( cls ` K ) ` ( S \ { x } ) ) <-> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) /\ x e. Y ) ) )
10		simp1 1061	. . . . . . . 8 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> J e. Top )
11	3	toptopon 20722	. . . . . . . 8 |- ( J e. Top <-> J e. (TopOn ` X ) )
12	10, 11	sylib 208	. . . . . . 7 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> J e. (TopOn ` X ) )
13		simp2 1062	. . . . . . 7 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> Y C_ X )
14		resttopon 20965	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ Y C_ X ) -> ( J ↾t Y ) e. (TopOn ` Y ) )
15	12, 13, 14	syl2anc 693	. . . . . 6 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( J ↾t Y ) e. (TopOn ` Y ) )
16	4, 15	syl5eqel 2705	. . . . 5 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> K e. (TopOn ` Y ) )
17		topontop 20718	. . . . 5 |- ( K e. (TopOn ` Y ) -> K e. Top )
18	16, 17	syl 17	. . . 4 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> K e. Top )
19		toponuni 20719	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y = U. K )
20	16, 19	syl 17	. . . . 5 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> Y = U. K )
21	1, 20	sseqtrd 3641	. . . 4 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> S C_ U. K )
22		eqid 2622	. . . . 5 |- U. K = U. K
23	22	islp 20944	. . . 4 |- ( ( K e. Top /\ S C_ U. K ) -> ( x e. ( ( limPt ` K ) ` S ) <-> x e. ( ( cls ` K ) ` ( S \ { x } ) ) ) )
24	18, 21, 23	syl2anc 693	. . 3 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( limPt ` K ) ` S ) <-> x e. ( ( cls ` K ) ` ( S \ { x } ) ) ) )
25		elin 3796	. . . 4 |- ( x e. ( ( ( limPt ` J ) ` S ) i^i Y ) <-> ( x e. ( ( limPt ` J ) ` S ) /\ x e. Y ) )
26	1, 13	sstrd 3613	. . . . . 6 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> S C_ X )
27	3	islp 20944	. . . . . 6 |- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( limPt ` J ) ` S ) <-> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) )
28	10, 26, 27	syl2anc 693	. . . . 5 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( limPt ` J ) ` S ) <-> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) )
29	28	anbi1d 741	. . . 4 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( ( x e. ( ( limPt ` J ) ` S ) /\ x e. Y ) <-> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) /\ x e. Y ) ) )
30	25, 29	syl5bb 272	. . 3 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( ( limPt ` J ) ` S ) i^i Y ) <-> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) /\ x e. Y ) ) )
31	9, 24, 30	3bitr4d 300	. 2 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( x e. ( ( limPt ` K ) ` S ) <-> x e. ( ( ( limPt ` J ) ` S ) i^i Y ) ) )
32	31	eqrdv 2620	1 |- ( ( J e. Top /\ Y C_ X /\ S C_ Y ) -> ( ( limPt ` K ) ` S ) = ( ( ( limPt ` J ) ` S ) i^i Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   clsccl 20822   limPtclp 20938

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cls 20825  df-lp 20940

This theorem is referenced by:  restperf  20988  lptioo2cn  39877  lptioo1cn  39878  limclner  39883  fourierdlem42  40366