Theorem nllyeq 23628

Description: Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyeq	⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Proof of Theorem nllyeq

Dummy variables 𝑗 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2852	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	rexbidv 3189	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵))
3	2	2ralbidv 3229	. . 3 ⊢ (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))
4	3	rabbidv 3423	. 2 ⊢ (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})
5		df-nlly 23624	. 2 ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
6		df-nlly 23624	. 2 ⊢ 𝑛-Locally 𝐵 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐵}
7	4, 5, 6	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416   ∩ cin 3904  𝒫 cpw 4562  {csn 4589  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17468  Topctop 23050  neicnei 23254  𝑛-Locally cnlly 23622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-nlly 23624

This theorem is referenced by: (None)