Theorem isnlly 23498

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 6920	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
2	1	fveq1d 6922	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑦}) = ((nei‘𝐽)‘{𝑦}))
3	2	ineq1d 4240	. . . . 5 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
4		oveq1 7455	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑢) = (𝐽 ↾t 𝑢))
5	4	eleq1d 2829	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝐽 ↾t 𝑢) ∈ 𝐴))
6	3, 5	rexeqbidv 3355	. . . 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 3184	. . 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 3346	. 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 23496	. 2 ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
10	8, 9	elrab2 3711	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975  𝒫 cpw 4622  {csn 4648  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  neicnei 23126  𝑛-Locally cnlly 23494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-nlly 23496

This theorem is referenced by:  nllytop  23502  nllyi  23504  llynlly  23506  nllyss  23509  nllyrest  23515  nllyidm  23518  hausllycmp  23523  cldllycmp  23524  txnlly  23666  cnllycmp  25007