Theorem llyeq 23426

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

Assertion

Ref	Expression
llyeq	⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Proof of Theorem llyeq

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

Step	Hyp	Ref	Expression
1		eleq2 2826	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	anbi2d 631	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
3	2	rexbidv 3162	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
4	3	2ralbidv 3202	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
5	4	rabbidv 3408	. 2 ⊢ (𝐴 = 𝐵 → {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)})
6		df-lly 23422	. 2 ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)}
7		df-lly 23422	. 2 ⊢ Locally 𝐵 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)}
8	5, 6, 7	3eqtr4g 2797	1 ⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401   ∩ cin 3902  𝒫 cpw 4556  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  Locally clly 23420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-lly 23422

This theorem is referenced by:  ismntoplly  34203