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Theorem neiin 32664
 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 471 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2 simpl 468 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3 eqid 2771 . . . . . . . . 9 𝐽 = 𝐽
43neiss2 21126 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 𝐽)
53neii1 21131 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 𝐽)
63neiint 21129 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑀 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
72, 4, 5, 6syl3anc 1476 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
81, 7mpbid 222 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9 ssinss1 3990 . . . . . 6 (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
108, 9syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
11103adant3 1126 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
12 inss2 3982 . . . . 5 (𝐴𝐵) ⊆ 𝐵
13 simpr 471 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14 simpl 468 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
153neiss2 21126 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 𝐽)
163neii1 21131 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
173neiint 21129 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1814, 15, 16, 17syl3anc 1476 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1913, 18mpbid 222 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20193adant2 1125 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
2112, 20syl5ss 3763 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑁))
2211, 21ssind 3985 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23 simp1 1130 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
2453adant3 1126 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 𝐽)
25163adant2 1125 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
263ntrin 21086 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 𝐽𝑁 𝐽) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2723, 24, 25, 26syl3anc 1476 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2822, 27sseqtr4d 3791 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁)))
29 ssinss1 3990 . . . . 5 (𝐴 𝐽 → (𝐴𝐵) ⊆ 𝐽)
304, 29syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ 𝐽)
31 ssinss1 3990 . . . . 5 (𝑀 𝐽 → (𝑀𝑁) ⊆ 𝐽)
325, 31syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀𝑁) ⊆ 𝐽)
333neiint 21129 . . . 4 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽 ∧ (𝑀𝑁) ⊆ 𝐽) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
342, 30, 32, 33syl3anc 1476 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
35343adant3 1126 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
3628, 35mpbird 247 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 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ∩ cin 3722   ⊆ wss 3723  ∪ cuni 4574  ‘cfv 6031  Topctop 20918  intcnt 21042  neicnei 21122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20919  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123 This theorem is referenced by: (None)
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