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Theorem neiint 22607
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 22606 . . . 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 1132 . . 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 1329 . 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 779 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))) β†’ 𝑆 βŠ† 𝑣)
61ssntr 22561 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
763adantl2 1167 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
87adantrrl 722 . . . . 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 3992 . . . 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 3156 . . 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 1191 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝐽 ∈ Top)
12 simpl3 1193 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝑁 βŠ† 𝑋)
131ntropn 22552 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
1411, 12, 13syl2anc 584 . . . . 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 485 . . . . 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 22560 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
1711, 12, 16syl2anc 584 . . . . 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 4008 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑆 βŠ† 𝑣 ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
19 sseq1 4007 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑣 βŠ† 𝑁 ↔ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁))
2018, 19anbi12d 631 . . . . . 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 3612 . . . . 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 835 . . . 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 413 . . 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 278 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22394  intcnt 22520  neicnei 22600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-ntr 22523  df-nei 22601
This theorem is referenced by:  opnnei  22623  topssnei  22627  iscnp4  22766  llycmpkgen2  23053  flimopn  23478  fclsneii  23520  fcfnei  23538  limcflf  25397  neiin  35212  cnneiima  47539
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