Theorem neiint 12939

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 12938	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
3	2	3adant3 1012	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
4	3	3anibar 1160	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
5		simprrl 534	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ 𝑣)
6	1	ssntr 12916	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
7	6	3adantl2 1149	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
8	7	adantrrl 483	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
9	5, 8	sstrd 3157	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
10	9	rexlimdvaa 2588	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11		simpl1 995	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12		simpl3 997	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁 ⊆ 𝑋)
13	1	ntropn 12911	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	11, 12, 13	syl2anc 409	. . . . 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 12915	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
17	11, 12, 16	syl2anc 409	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18		sseq2 3171	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆 ⊆ 𝑣 ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19		sseq1 3170	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣 ⊆ 𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
20	18, 19	anbi12d 470	. . . . . 6 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
21	20	rspcev 2834	. . . . 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 1231	. . . 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 973   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ⊆ wss 3121  ∪ cuni 3796  ‘cfv 5198  Topctop 12789  intcnt 12887  neicnei 12932

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-ntr 12890  df-nei 12933

This theorem is referenced by:  topssnei  12956  iscnp4  13012