Theorem nllyss 23540

Description: The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyss	⊢ (𝐴 ⊆ 𝐵 → 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.

Step	Hyp	Ref	Expression
1		ssel 3930	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 → (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	reximdv 3177	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 → ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵))
3	2	ralimdv 3176	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 → ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵))
4	3	ralimdv 3176	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵))
5	4	anim2d 621	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴) → (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵)))
6		isnlly 23529	. . 3 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴))
7		isnlly 23529	. . 3 ⊢ (𝑗 ∈ 𝑛-Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵))
8	5, 6, 7	3imtr4g 298	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑗 ∈ 𝑛-Locally 𝐴 → 𝑗 ∈ 𝑛-Locally 𝐵))
9	8	ssrdv 3942	1 ⊢ (𝐴 ⊆ 𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4555  {csn 4582  ‘cfv 6521  (class class class)co 7396   ↾_t crest 17449  Topctop 22953  neicnei 23157  𝑛-Locally cnlly 23525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399  df-nlly 23527

This theorem is referenced by:  iinllyconn  35604  cvmlift3  35678