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Theorem neiint 13684
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 13683 . . . 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 1017 . . 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 1165 . 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 539 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑁))) β†’ 𝑆 βŠ† 𝑣)
61ssntr 13661 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
763adantl2 1154 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 βŠ† 𝑁)) β†’ 𝑣 βŠ† ((intβ€˜π½)β€˜π‘))
87adantrrl 486 . . . . 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 3167 . . . 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 2595 . . 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 1000 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝐽 ∈ Top)
12 simpl3 1002 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑁 βŠ† 𝑋) ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)) β†’ 𝑁 βŠ† 𝑋)
131ntropn 13656 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) ∈ 𝐽)
1411, 12, 13syl2anc 411 . . . . 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 110 . . . . 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 13660 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁)
1711, 12, 16syl2anc 411 . . . . 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 3181 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑆 βŠ† 𝑣 ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘)))
19 sseq1 3180 . . . . . . 7 (𝑣 = ((intβ€˜π½)β€˜π‘) β†’ (𝑣 βŠ† 𝑁 ↔ ((intβ€˜π½)β€˜π‘) βŠ† 𝑁))
2018, 19anbi12d 473 . . . . . 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 2843 . . . . 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 1236 . . . 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 115 . . 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 129 . 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 188 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13536  intcnt 13632  neicnei 13677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13537  df-ntr 13635  df-nei 13678
This theorem is referenced by:  topssnei  13701  iscnp4  13757
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