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Theorem nllyss 22631
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 3914 . . . . . . 7 (𝐴𝐵 → ((𝑗t 𝑢) ∈ 𝐴 → (𝑗t 𝑢) ∈ 𝐵))
21reximdv 3202 . . . . . 6 (𝐴𝐵 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
32ralimdv 3109 . . . . 5 (𝐴𝐵 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
43ralimdv 3109 . . . 4 (𝐴𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
54anim2d 612 . . 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 22620 . . 3 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴))
7 isnlly 22620 . . 3 (𝑗 ∈ 𝑛-Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
85, 6, 73imtr4g 296 . 2 (𝐴𝐵 → (𝑗 ∈ 𝑛-Locally 𝐴𝑗 ∈ 𝑛-Locally 𝐵))
98ssrdv 3927 1 (𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3064  wrex 3065  cin 3886  wss 3887  𝒫 cpw 4533  {csn 4561  cfv 6433  (class class class)co 7275  t crest 17131  Topctop 22042  neicnei 22248  𝑛-Locally cnlly 22616
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-nlly 22618
This theorem is referenced by:  iinllyconn  33216  cvmlift3  33290
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