Theorem resttopon 14885

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 14728	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	adantr 276	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> J e. Top )
3		id 19	. . . 4 |- ( A C_ X -> A C_ X )
4		toponmax 14739	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
5		ssexg 4226	. . . 4 |- ( ( A C_ X /\ X e. J ) -> A e. _V )
6	3, 4, 5	syl2anr 290	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. _V )
7		resttop 14884	. . 3 |- ( ( J e. Top /\ A e. _V ) -> ( J ↾t A ) e. Top )
8	2, 6, 7	syl2anc 411	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. Top )
9		simpr 110	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A C_ X )
10		sseqin2 3424	. . . . . 6 |- ( A C_ X <-> ( X i^i A ) = A )
11	9, 10	sylib 122	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) = A )
12		simpl 109	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> J e. (TopOn ` X ) )
13	4	adantr 276	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> X e. J )
14		elrestr 13320	. . . . . 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 1271	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) e. ( J ↾t A ) )
16	11, 15	eqeltrrd 2307	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. ( J ↾t A ) )
17		elssuni 3919	. . . 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 13318	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
20	6, 19	syldan 282	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
21		inss2 3426	. . . . . . . . 9 |- ( x i^i A ) C_ A
22		vex 2803	. . . . . . . . . . 11 |- x e. _V
23	22	inex1 4221	. . . . . . . . . 10 |- ( x i^i A ) e. _V
24	23	elpw 3656	. . . . . . . . 9 |- ( ( x i^i A ) e. ~P A <-> ( x i^i A ) C_ A )
25	21, 24	mpbir 146	. . . . . . . 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 5798	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( x e. J |-> ( x i^i A ) ) : J --> ~P A )
28	27	frnd 5489	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ran ( x e. J |-> ( x i^i A ) ) C_ ~P A )
29	20, 28	eqsstrd 3261	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) C_ ~P A )
30		sspwuni 4053	. . . 4 |- ( ( J ↾t A ) C_ ~P A <-> U. ( J ↾t A ) C_ A )
31	29, 30	sylib 122	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> U. ( J ↾t A ) C_ A )
32	18, 31	eqssd 3242	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A = U. ( J ↾t A ) )
33		istopon 14727	. 2 |- ( ( J ↾t A ) e. (TopOn ` A ) <-> ( ( J ↾t A ) e. Top /\ A = U. ( J ↾t A ) ) )
34	8, 32, 33	sylanbrc 417	1 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   i^i cin 3197   C_ wss 3198   ~Pcpw 3650   U.cuni 3891   |-> cmpt 4148   ran crn 4724   ` cfv 5324  (class class class)co 6013   ↾_t crest 13312   Topctop 14711  TopOnctopon 14724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-rest 13314  df-topgen 13333  df-top 14712  df-topon 14725  df-bases 14757

This theorem is referenced by:  restuni  14886  stoig  14887  cnrest  14949  cnrest2  14950  cnrest2r  14951  cnptopresti  14952  cnptoprest  14953  cnptoprest2  14954  divcnap  15279  cncfmpt2fcntop  15313  cnplimcim  15381  cnlimcim  15385  cnlimc  15386  limccnpcntop  15389  limccnp2lem  15390  limccnp2cntop  15391  dvfvalap  15395  dvbss  15399  dvfgg  15402  dvcnp2cntop  15413  dvcn  15414  dvaddxxbr  15415  dvmulxxbr  15416  dvmptfsum  15439