Theorem isnlly 21494

Description: The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
isnlly	⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))

Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐽,𝑥,𝑦

Proof of Theorem isnlly

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6353	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
2	1	fveq1d 6355	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑦}) = ((nei‘𝐽)‘{𝑦}))
3	2	ineq1d 3956	. . . . 5 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
4		oveq1 6821	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑢) = (𝐽 ↾t 𝑢))
5	4	eleq1d 2824	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝐽 ↾t 𝑢) ∈ 𝐴))
6	3, 5	rexeqbidv 3292	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
7	6	ralbidv 3124	. . 3 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
8	7	raleqbi1dv 3285	. 2 ⊢ (𝑗 = 𝐽 → (∀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) ∈ 𝐴))
9		df-nlly 21492	. 2 ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
10	8, 9	elrab2 3507	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∩ cin 3714  𝒫 cpw 4302  {csn 4321  ‘cfv 6049  (class class class)co 6814   ↾_t crest 16303  Topctop 20920  neicnei 21123  𝑛-Locally cnlly 21490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-nlly 21492

This theorem is referenced by:  nllytop  21498  nllyi  21500  llynlly  21502  nllyss  21505  nllyrest  21511  nllyidm  21514  hausllycmp  21519  cldllycmp  21520  txnlly  21662  cnllycmp  22976