Theorem resttopon 12122

Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
resttopon	|- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )

Proof of Theorem resttopon

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 11963	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	adantr 272	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> J e. Top )
3		id 19	. . . 4 |- ( A C_ X -> A C_ X )
4		toponmax 11974	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
5		ssexg 4007	. . . 4 |- ( ( A C_ X /\ X e. J ) -> A e. _V )
6	3, 4, 5	syl2anr 286	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. _V )
7		resttop 12121	. . 3 |- ( ( J e. Top /\ A e. _V ) -> ( J ↾t A ) e. Top )
8	2, 6, 7	syl2anc 406	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. Top )
9		simpr 109	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A C_ X )
10		sseqin2 3242	. . . . . 6 |- ( A C_ X <-> ( X i^i A ) = A )
11	9, 10	sylib 121	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) = A )
12		simpl 108	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> J e. (TopOn ` X ) )
13	4	adantr 272	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> X e. J )
14		elrestr 11910	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. _V /\ X e. J ) -> ( X i^i A ) e. ( J ↾t A ) )
15	12, 6, 13, 14	syl3anc 1184	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) e. ( J ↾t A ) )
16	11, 15	eqeltrrd 2177	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. ( J ↾t A ) )
17		elssuni 3711	. . . 4 |- ( A e. ( J ↾t A ) -> A C_ U. ( J ↾t A ) )
18	16, 17	syl 14	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A C_ U. ( J ↾t A ) )
19		restval 11908	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
20	6, 19	syldan 278	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
21		inss2 3244	. . . . . . . . 9 |- ( x i^i A ) C_ A
22		vex 2644	. . . . . . . . . . 11 |- x e. _V
23	22	inex1 4002	. . . . . . . . . 10 |- ( x i^i A ) e. _V
24	23	elpw 3463	. . . . . . . . 9 |- ( ( x i^i A ) e. ~P A <-> ( x i^i A ) C_ A )
25	21, 24	mpbir 145	. . . . . . . 8 |- ( x i^i A ) e. ~P A
26	25	a1i 9	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ x e. J ) -> ( x i^i A ) e. ~P A )
27	26	fmpttd 5507	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( x e. J |-> ( x i^i A ) ) : J --> ~P A )
28	27	frnd 5218	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ran ( x e. J |-> ( x i^i A ) ) C_ ~P A )
29	20, 28	eqsstrd 3083	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) C_ ~P A )
30		sspwuni 3843	. . . 4 |- ( ( J ↾t A ) C_ ~P A <-> U. ( J ↾t A ) C_ A )
31	29, 30	sylib 121	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> U. ( J ↾t A ) C_ A )
32	18, 31	eqssd 3064	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A = U. ( J ↾t A ) )
33		istopon 11962	. 2 |- ( ( J ↾t A ) e. (TopOn ` A ) <-> ( ( J ↾t A ) e. Top /\ A = U. ( J ↾t A ) ) )
34	8, 32, 33	sylanbrc 411	1 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   _Vcvv 2641   i^i cin 3020   C_ wss 3021   ~Pcpw 3457   U.cuni 3683   |-> cmpt 3929   ran crn 4478   ` cfv 5059  (class class class)co 5706   ↾_t crest 11902   Topctop 11946  TopOnctopon 11959

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-rest 11904  df-topgen 11923  df-top 11947  df-topon 11960  df-bases 11992

This theorem is referenced by:  restuni  12123  stoig  12124  cnrest  12185  cnrest2  12186  cnrest2r  12187  cnptopresti  12188  cnptoprest  12189  cnptoprest2  12190  cnplimcim  12516  cnlimcim  12517  limccnpcntop  12520  dvfvalap  12523  dvbss  12527  dvfgg  12530