Theorem llyss 22087

Description: The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llyss	⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Proof of Theorem llyss

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

Step	Hyp	Ref	Expression
1		ssel 3961	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 → (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	anim2d 613	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
3	2	reximdv 3273	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
4	3	ralimdv 3178	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) → ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
5	4	ralimdv 3178	. . . 4 ⊢ (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)))
6	5	anim2d 613	. . 3 ⊢ (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))))
7		islly 22076	. . 3 ⊢ (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)))
8		islly 22076	. . 3 ⊢ (𝑗 ∈ Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
9	6, 7, 8	3imtr4g 298	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Locally 𝐵))
10	9	ssrdv 3973	1 ⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  (class class class)co 7156   ↾_t crest 16694  Topctop 21501  Locally clly 22072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-lly 22074

This theorem is referenced by: (None)