MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  topssnei Structured version   Visualization version   GIF version
Theorem topssnei 23056
Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypotheses
Ref Expression
tpnei.1 𝑋 = βˆͺ 𝐽
topssnei.2 π‘Œ = βˆͺ 𝐾
Assertion
Ref Expression
topssnei (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ 𝐽 βŠ† 𝐾) β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† ((neiβ€˜πΎ)β€˜π‘†))
Proof of Theorem topssnei
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simpl2 1189 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐾 ∈ Top)
2 simprl 769 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 βŠ† 𝐾)
3 simpl1 1188 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 ∈ Top)
4 simprr 771 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
5 tpnei.1 . . . . . . . . 9 𝑋 = βˆͺ 𝐽
65neii1 23038 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ π‘₯ βŠ† 𝑋)
73, 4, 6syl2anc 582 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ βŠ† 𝑋)
85ntropn 22981 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
93, 7, 8syl2anc 582 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
102, 9sseldd 3983 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐾)
115neiss2 23033 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
123, 4, 11syl2anc 582 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑆 βŠ† 𝑋)
135neiint 23036 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ π‘₯ βŠ† 𝑋) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘₯)))
143, 12, 7, 13syl3anc 1368 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ↔ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘₯)))
154, 14mpbid 231 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘₯))
16 opnneiss 23050 . . . . 5 ((𝐾 ∈ Top ∧ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐾 ∧ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘₯)) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ ((neiβ€˜πΎ)β€˜π‘†))
171, 10, 15, 16syl3anc 1368 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ ((neiβ€˜πΎ)β€˜π‘†))
185ntrss2 22989 . . . . 5 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
193, 7, 18syl2anc 582 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
20 simpl3 1190 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑋 = π‘Œ)
217, 20sseqtrd 4022 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ βŠ† π‘Œ)
22 topssnei.2 . . . . 5 π‘Œ = βˆͺ 𝐾
2322ssnei2 23048 . . . 4 (((𝐾 ∈ Top ∧ ((intβ€˜π½)β€˜π‘₯) ∈ ((neiβ€˜πΎ)β€˜π‘†)) ∧ (((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯ ∧ π‘₯ βŠ† π‘Œ)) β†’ π‘₯ ∈ ((neiβ€˜πΎ)β€˜π‘†))
241, 17, 19, 21, 23syl22anc 837 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ ∈ ((neiβ€˜πΎ)β€˜π‘†))
2524expr 455 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) β†’ π‘₯ ∈ ((neiβ€˜πΎ)β€˜π‘†)))
2625ssrdv 3988 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ 𝐽 βŠ† 𝐾) β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† ((neiβ€˜πΎ)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  βˆͺ cuni 4912  β€˜cfv 6553  Topctop 22823  intcnt 22949  neicnei 23029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-top 22824  df-ntr 22952  df-nei 23030
This theorem is referenced by:  flimss1  23905
  Copyright terms: Public domain W3C validator