Theorem neiint 14324

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 14323	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
3	2	3adant3 1019	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
4	3	3anibar 1167	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
5		simprrl 539	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ 𝑣)
6	1	ssntr 14301	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
7	6	3adantl2 1156	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
8	7	adantrrl 486	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
9	5, 8	sstrd 3190	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
10	9	rexlimdvaa 2612	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11		simpl1 1002	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12		simpl3 1004	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁 ⊆ 𝑋)
13	1	ntropn 14296	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	11, 12, 13	syl2anc 411	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15		simpr 110	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
16	1	ntrss2 14300	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
17	11, 12, 16	syl2anc 411	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18		sseq2 3204	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆 ⊆ 𝑣 ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19		sseq1 3203	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣 ⊆ 𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
20	18, 19	anbi12d 473	. . . . . 6 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
21	20	rspcev 2865	. . . . 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 1247	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))
23	22	ex 115	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
24	10, 23	impbid 129	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
25	4, 24	bitrd 188	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3154  ∪ cuni 3836  ‘cfv 5255  Topctop 14176  intcnt 14272  neicnei 14317

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 14177  df-ntr 14275  df-nei 14318

This theorem is referenced by:  topssnei  14341  iscnp4  14397