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Theorem topssnei 22983
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 768 . . . . . 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 770 . . . . . . . 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 22965 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ π‘₯ βŠ† 𝑋)
73, 4, 6syl2anc 583 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ π‘₯ βŠ† 𝑋)
85ntropn 22908 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) ∈ 𝐽)
93, 7, 8syl2anc 583 . . . . . 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 3978 . . . . 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 22960 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
123, 4, 11syl2anc 583 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = π‘Œ) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ 𝑆 βŠ† 𝑋)
135neiint 22963 . . . . . . 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 22977 . . . . 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 22916 . . . . 5 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘₯) βŠ† π‘₯)
193, 7, 18syl2anc 583 . . . 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 4017 . . . 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 22975 . . . 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 836 . . 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 456 . 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 3983 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6537  Topctop 22750  intcnt 22876  neicnei 22956
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 22751  df-ntr 22879  df-nei 22957
This theorem is referenced by:  flimss1  23832
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