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Theorem topssnei 22848
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 1190 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐾 ∈ Top)
2 simprl 767 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 βŠ† 𝐾)
3 simpl1 1189 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝐽 ∈ Top)
4 simprr 769 . . . . . . . 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 22830 . . . . . . . 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 22773 . . . . . . 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 3982 . . . . 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 22825 . . . . . . . 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 22828 . . . . . . 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 1369 . . . . . 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 22842 . . . . 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 1369 . . . 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 22781 . . . . 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 1191 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑋 = π‘Œ)
217, 20sseqtrd 4021 . . . 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 22840 . . . 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 835 . . 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 3987 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 1085   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6542  Topctop 22615  intcnt 22741  neicnei 22821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22616  df-ntr 22744  df-nei 22822
This theorem is referenced by:  flimss1  23697
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