Theorem llyeq 21602

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 2867	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	anbi2d 623	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
3	2	rexbidv 3233	. . . 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 3170	. . 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 3373	. 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 21598	. 2 ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)}
7		df-lly 21598	. 2 ⊢ Locally 𝐵 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)}
8	5, 6, 7	3eqtr4g 2858	1 ⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3089  ∃wrex 3090  {crab 3093   ∩ cin 3768  𝒫 cpw 4349  (class class class)co 6878   ↾_t crest 16396  Topctop 21026  Locally clly 21596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-rex 3095  df-rab 3098  df-lly 21598

This theorem is referenced by:  ismntoplly  30585