Theorem neiint 23048

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 23047	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
3	2	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))))
4	3	3anibar 1330	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁)))
5		simprrl 780	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ 𝑣)
6	1	ssntr 23002	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
7	6	3adantl2 1168	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
8	7	adantrrl 724	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
9	5, 8	sstrd 3944	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
10	9	rexlimdvaa 3138	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11		simpl1 1192	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12		simpl3 1194	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁 ⊆ 𝑋)
13	1	ntropn 22993	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
14	11, 12, 13	syl2anc 584	. . . . 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 23001	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ⊆ 𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
17	11, 12, 16	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18		sseq2 3960	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆 ⊆ 𝑣 ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19		sseq1 3959	. . . . . . 7 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣 ⊆ 𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
20	18, 19	anbi12d 632	. . . . . 6 ⊢ (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
21	20	rspcev 3576	. . . . 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 836	. . . 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 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ⊆ wss 3901  ∪ cuni 4863  ‘cfv 6492  Topctop 22837  intcnt 22961  neicnei 23041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-ntr 22964  df-nei 23042

This theorem is referenced by:  opnnei  23064  topssnei  23068  iscnp4  23207  llycmpkgen2  23494  flimopn  23919  fclsneii  23961  fcfnei  23979  limcflf  25838  neiin  36526  cnneiima  49158