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Theorem neiin 35217
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 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
Proof of Theorem neiin
StepHypRef Expression
1 simpr 486 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄))
2 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ 𝐽 ∈ Top)
3 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
43neiss2 22605 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
53neii1 22610 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ 𝑀 βŠ† βˆͺ 𝐽)
63neiint 22608 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 βŠ† βˆͺ 𝐽 ∧ 𝑀 βŠ† βˆͺ 𝐽) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ↔ 𝐴 βŠ† ((intβ€˜π½)β€˜π‘€)))
72, 4, 5, 6syl3anc 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ↔ 𝐴 βŠ† ((intβ€˜π½)β€˜π‘€)))
81, 7mpbid 231 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ 𝐴 βŠ† ((intβ€˜π½)β€˜π‘€))
9 ssinss1 4238 . . . . . 6 (𝐴 βŠ† ((intβ€˜π½)β€˜π‘€) β†’ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜π‘€))
108, 9syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜π‘€))
11103adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜π‘€))
12 inss2 4230 . . . . 5 (𝐴 ∩ 𝐡) βŠ† 𝐡
13 simpr 486 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅))
14 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝐽 ∈ Top)
153neiss2 22605 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝐡 βŠ† βˆͺ 𝐽)
163neii1 22610 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
173neiint 22608 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐡 βŠ† βˆͺ 𝐽 ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π΅) ↔ 𝐡 βŠ† ((intβ€˜π½)β€˜π‘)))
1814, 15, 16, 17syl3anc 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π΅) ↔ 𝐡 βŠ† ((intβ€˜π½)β€˜π‘)))
1913, 18mpbid 231 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝐡 βŠ† ((intβ€˜π½)β€˜π‘))
20193adant2 1132 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝐡 βŠ† ((intβ€˜π½)β€˜π‘))
2112, 20sstrid 3994 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜π‘))
2211, 21ssind 4233 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝐴 ∩ 𝐡) βŠ† (((intβ€˜π½)β€˜π‘€) ∩ ((intβ€˜π½)β€˜π‘)))
23 simp1 1137 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝐽 ∈ Top)
2453adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝑀 βŠ† βˆͺ 𝐽)
25163adant2 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
263ntrin 22565 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 βŠ† βˆͺ 𝐽 ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁)) = (((intβ€˜π½)β€˜π‘€) ∩ ((intβ€˜π½)β€˜π‘)))
2723, 24, 25, 26syl3anc 1372 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁)) = (((intβ€˜π½)β€˜π‘€) ∩ ((intβ€˜π½)β€˜π‘)))
2822, 27sseqtrrd 4024 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁)))
29 ssinss1 4238 . . . . 5 (𝐴 βŠ† βˆͺ 𝐽 β†’ (𝐴 ∩ 𝐡) βŠ† βˆͺ 𝐽)
304, 29syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ (𝐴 ∩ 𝐡) βŠ† βˆͺ 𝐽)
31 ssinss1 4238 . . . . 5 (𝑀 βŠ† βˆͺ 𝐽 β†’ (𝑀 ∩ 𝑁) βŠ† βˆͺ 𝐽)
325, 31syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ (𝑀 ∩ 𝑁) βŠ† βˆͺ 𝐽)
333neiint 22608 . . . 4 ((𝐽 ∈ Top ∧ (𝐴 ∩ 𝐡) βŠ† βˆͺ 𝐽 ∧ (𝑀 ∩ 𝑁) βŠ† βˆͺ 𝐽) β†’ ((𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)) ↔ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁))))
342, 30, 32, 33syl3anc 1372 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄)) β†’ ((𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)) ↔ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁))))
35343adant3 1133 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ ((𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)) ↔ (𝐴 ∩ 𝐡) βŠ† ((intβ€˜π½)β€˜(𝑀 ∩ 𝑁))))
3628, 35mpbird 257 1 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  intcnt 22521  neicnei 22601
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602
This theorem is referenced by: (None)
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