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Theorem nllyss 22074
Description: The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyss (𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)

Proof of Theorem nllyss
Dummy variables 𝑗 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3945 . . . . . . 7 (𝐴𝐵 → ((𝑗t 𝑢) ∈ 𝐴 → (𝑗t 𝑢) ∈ 𝐵))
21reximdv 3265 . . . . . 6 (𝐴𝐵 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
32ralimdv 3172 . . . . 5 (𝐴𝐵 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
43ralimdv 3172 . . . 4 (𝐴𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
54anim2d 614 . . 3 (𝐴𝐵 → ((𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴) → (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵)))
6 isnlly 22063 . . 3 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴))
7 isnlly 22063 . . 3 (𝑗 ∈ 𝑛-Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
85, 6, 73imtr4g 299 . 2 (𝐴𝐵 → (𝑗 ∈ 𝑛-Locally 𝐴𝑗 ∈ 𝑛-Locally 𝐵))
98ssrdv 3957 1 (𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3132  wrex 3133  cin 3917  wss 3918  𝒫 cpw 4520  {csn 4548  cfv 6336  (class class class)co 7138  t crest 16683  Topctop 21487  neicnei 21691  𝑛-Locally cnlly 22059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344  df-ov 7141  df-nlly 22061
This theorem is referenced by:  iinllyconn  32519  cvmlift3  32593
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