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Theorem topssnei 20738
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 1058 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2 simprl 790 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽𝐾)
3 simpl1 1057 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4 simprr 792 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5 tpnei.1 . . . . . . . . 9 𝑋 = 𝐽
65neii1 20720 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
73, 4, 6syl2anc 691 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥𝑋)
85ntropn 20663 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
93, 7, 8syl2anc 691 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
102, 9sseldd 3569 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
115neiss2 20715 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
123, 4, 11syl2anc 691 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆𝑋)
135neiint 20718 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑥𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
143, 12, 7, 13syl3anc 1318 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
154, 14mpbid 221 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16 opnneiss 20732 . . . . 5 ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
171, 10, 15, 16syl3anc 1318 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
185ntrss2 20671 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
193, 7, 18syl2anc 691 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20 simpl3 1059 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
217, 20sseqtrd 3604 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥𝑌)
22 topssnei.2 . . . . 5 𝑌 = 𝐾
2322ssnei2 20730 . . . 4 (((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆)) ∧ (((int‘𝐽)‘𝑥) ⊆ 𝑥𝑥𝑌)) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
241, 17, 19, 21, 23syl22anc 1319 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽𝐾𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
2524expr 641 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
2625ssrdv 3574 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540   cuni 4372  cfv 5804  Topctop 20517  intcnt 20631  neicnei 20711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-ntr 20634  df-nei 20712
This theorem is referenced by:  flimss1  21587
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