Theorem resttopon 14758

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 14601	. . . 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 14612	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
5		ssexg 4199	. . . 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 14757	. . 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 3400	. . . . . 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 13194	. . . . . 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 1250	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( X i^i A ) e. ( J ↾t A ) )
16	11, 15	eqeltrrd 2285	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. ( J ↾t A ) )
17		elssuni 3892	. . . 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 13192	. . . . . 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 3402	. . . . . . . . 9 |- ( x i^i A ) C_ A
22		vex 2779	. . . . . . . . . . 11 |- x e. _V
23	22	inex1 4194	. . . . . . . . . 10 |- ( x i^i A ) e. _V
24	23	elpw 3632	. . . . . . . . 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 5758	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( x e. J |-> ( x i^i A ) ) : J --> ~P A )
28	27	frnd 5455	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ran ( x e. J |-> ( x i^i A ) ) C_ ~P A )
29	20, 28	eqsstrd 3237	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) C_ ~P A )
30		sspwuni 4026	. . . 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 3218	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A = U. ( J ↾t A ) )
33		istopon 14600	. 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 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   C_ wss 3174   ~Pcpw 3626   U.cuni 3864   |-> cmpt 4121   ran crn 4694   ` cfv 5290  (class class class)co 5967   ↾_t crest 13186   Topctop 14584  TopOnctopon 14597

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-rest 13188  df-topgen 13207  df-top 14585  df-topon 14598  df-bases 14630

This theorem is referenced by:  restuni  14759  stoig  14760  cnrest  14822  cnrest2  14823  cnrest2r  14824  cnptopresti  14825  cnptoprest  14826  cnptoprest2  14827  divcnap  15152  cncfmpt2fcntop  15186  cnplimcim  15254  cnlimcim  15258  cnlimc  15259  limccnpcntop  15262  limccnp2lem  15263  limccnp2cntop  15264  dvfvalap  15268  dvbss  15272  dvfgg  15275  dvcnp2cntop  15286  dvcn  15287  dvaddxxbr  15288  dvmulxxbr  15289  dvmptfsum  15312