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Theorem neiint 22608
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 = βˆͺ 𝐽
Assertion
Ref Expression
neiint ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
Proof of Theorem neiint
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = βˆͺ 𝐽
21isnei 22607 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))))
323adant3 1133 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))))
433anibar 1330 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁)))
5 simprrl 780 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))) β†’ 𝑆 βŠ† 𝑣)
61ssntr 22562 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
763adantl2 1168 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
87adantrrl 723 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
95, 8sstrd 3993 . . . 4 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘))
109rexlimdvaa 3157 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
11 simpl1 1192 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝐽 ∈ Top)
12 simpl3 1194 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝑁 βŠ† 𝑋)
131ntropn 22553 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
1411, 12, 13syl2anc 585 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
15 simpr 486 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘))
161ntrss2 22561 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
1711, 12, 16syl2anc 585 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
18 sseq2 4009 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑆 βŠ† 𝑣 ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
19 sseq1 4008 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑣 βŠ† 𝑁 ↔ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁))
2018, 19anbi12d 632 . . . . . 6 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ ((𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁) ↔ (𝑆 βŠ† ((intβ€˜π½)β€˜π‘) ∧ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)))
2120rspcev 3613 . . . . 5 ((((intβ€˜π½)β€˜π‘) ∈ 𝐽 ∧ (𝑆 βŠ† ((intβ€˜π½)β€˜π‘) ∧ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))
2214, 15, 17, 21syl12anc 836 . . . 4 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))
2322ex 414 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑆 βŠ† ((intβ€˜π½)β€˜π‘) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁)))
2410, 23impbid 211 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁) ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
254, 24bitrd 279 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   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-iun 5000  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-ntr 22524  df-nei 22602
This theorem is referenced by:  opnnei  22624  topssnei  22628  iscnp4  22767  llycmpkgen2  23054  flimopn  23479  fclsneii  23521  fcfnei  23539  limcflf  25398  neiin  35217  cnneiima  47549
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