Theorem resttopon 15363

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 15206	. . . 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 15217	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
5		ssexg 4272	. . . 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 15362	. . 3 |- ( ( J e. Top /\ A e. _V ) -> ( J ↾t A ) e. Top )
8	2, 6, 7	syl2anc 415	. 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 3450	. . . . . 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 13654	. . . . . 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 1278	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) e. ( J ↾t A ) )
16	11, 15	eqeltrrd 2316	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. ( J ↾t A ) )
17		elssuni 3963	. . . 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 13652	. . . . . 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 3452	. . . . . . . . 9 |- ( x i^i A ) C_ A
22		vex 2824	. . . . . . . . . . 11 |- x e. _V
23	22	inex1 4267	. . . . . . . . . 10 |- ( x i^i A ) e. _V
24	23	elpw 3694	. . . . . . . . 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 5863	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( x e. J |-> ( x i^i A ) ) : J --> ~P A )
28	27	frnd 5543	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ran ( x e. J |-> ( x i^i A ) ) C_ ~P A )
29	20, 28	eqsstrd 3284	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) C_ ~P A )
30		sspwuni 4097	. . . 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 3265	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A = U. ( J ↾t A ) )
33		istopon 15205	. 2 |- ( ( J ↾t A ) e. (TopOn ` A ) <-> ( ( J ↾t A ) e. Top /\ A = U. ( J ↾t A ) ) )
34	8, 32, 33	sylanbrc 421	1 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   U.cuni 3935   |-> cmpt 4192   ran crn 4775   ` cfv 5377  (class class class)co 6085   ↾_t crest 13646   Topctop 15189  TopOnctopon 15202

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-rest 13648  df-topgen 13667  df-top 15190  df-topon 15203  df-bases 15235

This theorem is used by:  restuni  15364  stoig  15365  cnrest  15427  cnrest2  15428  cnrest2r  15429  cnptopresti  15430  cnptoprest  15431  cnptoprest2  15432  divcnap  15757  cncfmpt2fcntop  15791  cnplimcim  15859  cnlimcim  15863  cnlimc  15864  limccnpcntop  15867  limccnp2lem  15868  limccnp2cntop  15869  dvfvalap  15873  dvbss  15877  dvfgg  15880  dvcnp2cntop  15891  dvcn  15892  dvaddxxbr  15893  dvmulxxbr  15894  dvmptfsum  15917