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Theorem neiin 35155
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 486 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3 eqid 2733 . . . . . . . . 9 𝐽 = 𝐽
43neiss2 22587 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 𝐽)
53neii1 22592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 𝐽)
63neiint 22590 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑀 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
72, 4, 5, 6syl3anc 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
81, 7mpbid 231 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9 ssinss1 4236 . . . . . 6 (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
108, 9syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
11103adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
12 inss2 4228 . . . . 5 (𝐴𝐵) ⊆ 𝐵
13 simpr 486 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
153neiss2 22587 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 𝐽)
163neii1 22592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
173neiint 22590 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1814, 15, 16, 17syl3anc 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1913, 18mpbid 231 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20193adant2 1132 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
2112, 20sstrid 3992 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑁))
2211, 21ssind 4231 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23 simp1 1137 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
2453adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 𝐽)
25163adant2 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
263ntrin 22547 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 𝐽𝑁 𝐽) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2723, 24, 25, 26syl3anc 1372 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2822, 27sseqtrrd 4022 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁)))
29 ssinss1 4236 . . . . 5 (𝐴 𝐽 → (𝐴𝐵) ⊆ 𝐽)
304, 29syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ 𝐽)
31 ssinss1 4236 . . . . 5 (𝑀 𝐽 → (𝑀𝑁) ⊆ 𝐽)
325, 31syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀𝑁) ⊆ 𝐽)
333neiint 22590 . . . 4 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽 ∧ (𝑀𝑁) ⊆ 𝐽) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
342, 30, 32, 33syl3anc 1372 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
35343adant3 1133 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
3628, 35mpbird 257 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cin 3946  wss 3947   cuni 4907  cfv 6540  Topctop 22377  intcnt 22503  neicnei 22583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22378  df-cld 22505  df-ntr 22506  df-cls 22507  df-nei 22584
This theorem is referenced by: (None)
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