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Theorem neiin 33577
Description: Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
Assertion
Ref Expression
neiin ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))

Proof of Theorem neiin
StepHypRef Expression
1 simpr 485 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2 simpl 483 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3 eqid 2818 . . . . . . . . 9 𝐽 = 𝐽
43neiss2 21637 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 𝐽)
53neii1 21642 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 𝐽)
63neiint 21640 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑀 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
72, 4, 5, 6syl3anc 1363 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
81, 7mpbid 233 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9 ssinss1 4211 . . . . . 6 (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
108, 9syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
11103adant3 1124 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
12 inss2 4203 . . . . 5 (𝐴𝐵) ⊆ 𝐵
13 simpr 485 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14 simpl 483 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
153neiss2 21637 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 𝐽)
163neii1 21642 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
173neiint 21640 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1814, 15, 16, 17syl3anc 1363 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1913, 18mpbid 233 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20193adant2 1123 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
2112, 20sstrid 3975 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑁))
2211, 21ssind 4206 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23 simp1 1128 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
2453adant3 1124 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 𝐽)
25163adant2 1123 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
263ntrin 21597 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 𝐽𝑁 𝐽) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2723, 24, 25, 26syl3anc 1363 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2822, 27sseqtrrd 4005 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁)))
29 ssinss1 4211 . . . . 5 (𝐴 𝐽 → (𝐴𝐵) ⊆ 𝐽)
304, 29syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ 𝐽)
31 ssinss1 4211 . . . . 5 (𝑀 𝐽 → (𝑀𝑁) ⊆ 𝐽)
325, 31syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀𝑁) ⊆ 𝐽)
333neiint 21640 . . . 4 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽 ∧ (𝑀𝑁) ⊆ 𝐽) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
342, 30, 32, 33syl3anc 1363 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
35343adant3 1124 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
3628, 35mpbird 258 1 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932  wss 3933   cuni 4830  cfv 6348  Topctop 21429  intcnt 21553  neicnei 21633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634
This theorem is referenced by: (None)
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