Theorem neiint 23069

Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
neifval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neiint	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Proof of Theorem neiint

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	isnei 23068	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
3	2	3adant3 1133	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
4	3	3anibar 1331	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
5		simprrl 781	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ 𝑣)
6	1	ssntr 23023	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
7	6	3adantl2 1169	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
8	7	adantrrl 725	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
9	5, 8	sstrd 3932	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
10	9	rexlimdvaa 3139	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11		simpl1 1193	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12		simpl3 1195	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁 ⊆ 𝑋)
13	1	ntropn 23014	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	11, 12, 13	syl2anc 585	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15		simpr 484	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
16	1	ntrss2 23022	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
17	11, 12, 16	syl2anc 585	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18		sseq2 3948	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆 ⊆ 𝑣 ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19		sseq1 3947	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣 ⊆ 𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
20	18, 19	anbi12d 633	. . . . . 6 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
21	20	rspcev 3564	. . . . 5 ⊢ ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))
22	14, 15, 17, 21	syl12anc 837	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))
23	22	ex 412	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
24	10, 23	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
25	4, 24	bitrd 279	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  intcnt 22982  neicnei 23062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-top 22859  df-ntr 22985  df-nei 23063

This theorem is referenced by:  opnnei  23085  topssnei  23089  iscnp4  23228  llycmpkgen2  23515  flimopn  23940  fclsneii  23982  fcfnei  24000  limcflf  25848  neiin  36514  cnneiima  49392