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Mirrors  >  Home  >  ILE Home  >  Th. List  >  topssnei GIF version
Theorem topssnei 13747
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 1001 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐾 ∈ Top)
2 simprl 529 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 βŠ† 𝐾)
3 simpl1 1000 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 ∈ Top)
4 simprr 531 . . . . . . . 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 13732 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ π‘₯ βŠ† 𝑋)
73, 4, 6syl2anc 411 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ βŠ† 𝑋)
85ntropn 13702 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
93, 7, 8syl2anc 411 . . . . . 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 3158 . . . . 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 13727 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
123, 4, 11syl2anc 411 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑆 βŠ† 𝑋)
135neiint 13730 . . . . . . 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 1238 . . . . . 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 147 . . . . 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 13743 . . . . 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 1238 . . . 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 13706 . . . . 5 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
193, 7, 18syl2anc 411 . . . 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 1002 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑋 = π‘Œ)
217, 20sseqtrd 3195 . . . 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 13742 . . . 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 1239 . . 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 375 . 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 3163 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13582  intcnt 13678  neicnei 13723
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 13583  df-ntr 13681  df-nei 13724
This theorem is referenced by: (None)
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