Theorem resttopon 12340

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 12181	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	adantr 274	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> J e. Top )
3		id 19	. . . 4 |- ( A C_ X -> A C_ X )
4		toponmax 12192	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
5		ssexg 4067	. . . 4 |- ( ( A C_ X /\ X e. J ) -> A e. _V )
6	3, 4, 5	syl2anr 288	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. _V )
7		resttop 12339	. . 3 |- ( ( J e. Top /\ A e. _V ) -> ( J ↾t A ) e. Top )
8	2, 6, 7	syl2anc 408	. 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 3295	. . . . . 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 274	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> X e. J )
14		elrestr 12128	. . . . . 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 1216	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) e. ( J ↾t A ) )
16	11, 15	eqeltrrd 2217	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. ( J ↾t A ) )
17		elssuni 3764	. . . 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 12126	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
20	6, 19	syldan 280	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
21		inss2 3297	. . . . . . . . 9 |- ( x i^i A ) C_ A
22		vex 2689	. . . . . . . . . . 11 |- x e. _V
23	22	inex1 4062	. . . . . . . . . 10 |- ( x i^i A ) e. _V
24	23	elpw 3516	. . . . . . . . 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 5575	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( x e. J |-> ( x i^i A ) ) : J --> ~P A )
28	27	frnd 5282	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ran ( x e. J |-> ( x i^i A ) ) C_ ~P A )
29	20, 28	eqsstrd 3133	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) C_ ~P A )
30		sspwuni 3897	. . . 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 3114	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A = U. ( J ↾t A ) )
33		istopon 12180	. 2 |- ( ( J ↾t A ) e. (TopOn ` A ) <-> ( ( J ↾t A ) e. Top /\ A = U. ( J ↾t A ) ) )
34	8, 32, 33	sylanbrc 413	1 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   i^i cin 3070   C_ wss 3071   ~Pcpw 3510   U.cuni 3736   |-> cmpt 3989   ran crn 4540   ` cfv 5123  (class class class)co 5774   ↾_t crest 12120   Topctop 12164  TopOnctopon 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-rest 12122  df-topgen 12141  df-top 12165  df-topon 12178  df-bases 12210

This theorem is referenced by:  restuni  12341  stoig  12342  cnrest  12404  cnrest2  12405  cnrest2r  12406  cnptopresti  12407  cnptoprest  12408  cnptoprest2  12409  divcnap  12724  cncfmpt2fcntop  12754  cnplimcim  12805  cnlimcim  12809  cnlimc  12810  limccnpcntop  12813  limccnp2lem  12814  limccnp2cntop  12815  dvfvalap  12819  dvbss  12823  dvfgg  12826  dvcnp2cntop  12832  dvcn  12833  dvaddxxbr  12834  dvmulxxbr  12835