Theorem neiint 12303

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 12302	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
3	2	3adant3 1001	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
4	3	3anibar 1149	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
5		simprrl 528	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ 𝑣)
6	1	ssntr 12280	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
7	6	3adantl2 1138	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
8	7	adantrrl 477	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
9	5, 8	sstrd 3102	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
10	9	rexlimdvaa 2548	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11		simpl1 984	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12		simpl3 986	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁 ⊆ 𝑋)
13	1	ntropn 12275	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	11, 12, 13	syl2anc 408	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15		simpr 109	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
16	1	ntrss2 12279	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
17	11, 12, 16	syl2anc 408	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18		sseq2 3116	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆 ⊆ 𝑣 ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19		sseq1 3115	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣 ⊆ 𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
20	18, 19	anbi12d 464	. . . . . 6 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
21	20	rspcev 2784	. . . . 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 1214	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))
23	22	ex 114	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
24	10, 23	impbid 128	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
25	4, 24	bitrd 187	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2415   ⊆ wss 3066  ∪ cuni 3731  ‘cfv 5118  Topctop 12153  intcnt 12251  neicnei 12296

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-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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-ntr 12254  df-nei 12297

This theorem is referenced by:  topssnei  12320  iscnp4  12376