Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  topssnei Structured version   Visualization version   GIF version

Theorem topssnei 21839
 Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypotheses
Ref Expression
tpnei.1 𝑋 = 𝐽
topssnei.2 𝑌 = 𝐾
Assertion
Ref Expression
topssnei (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Proof of Theorem topssnei
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1190 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2 simprl 770 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽𝐾)
3 simpl1 1189 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4 simprr 772 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5 tpnei.1 . . . . . . . . 9 𝑋 = 𝐽
65neii1 21821 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
73, 4, 6syl2anc 587 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥𝑋)
85ntropn 21764 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
93, 7, 8syl2anc 587 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
102, 9sseldd 3896 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
115neiss2 21816 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
123, 4, 11syl2anc 587 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆𝑋)
135neiint 21819 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑥𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
143, 12, 7, 13syl3anc 1369 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
154, 14mpbid 235 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16 opnneiss 21833 . . . . 5 ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
171, 10, 15, 16syl3anc 1369 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
185ntrss2 21772 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
193, 7, 18syl2anc 587 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20 simpl3 1191 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
217, 20sseqtrd 3935 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥𝑌)
22 topssnei.2 . . . . 5 𝑌 = 𝐾
2322ssnei2 21831 . . . 4 (((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆)) ∧ (((int‘𝐽)‘𝑥) ⊆ 𝑥𝑥𝑌)) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
241, 17, 19, 21, 23syl22anc 837 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
2524expr 460 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
2625ssrdv 3901 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ⊆ wss 3861  ∪ cuni 4802  ‘cfv 6341  Topctop 21608  intcnt 21732  neicnei 21812 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-top 21609  df-ntr 21735  df-nei 21813 This theorem is referenced by:  flimss1  22688
 Copyright terms: Public domain W3C validator