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Theorem llyss 21092
 Description: The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyss (𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Proof of Theorem llyss
Dummy variables 𝑗 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3562 . . . . . . . 8 (𝐴𝐵 → ((𝑗t 𝑢) ∈ 𝐴 → (𝑗t 𝑢) ∈ 𝐵))
21anim2d 587 . . . . . . 7 (𝐴𝐵 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
32reximdv 2999 . . . . . 6 (𝐴𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
43ralimdv 2946 . . . . 5 (𝐴𝐵 → (∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
54ralimdv 2946 . . . 4 (𝐴𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
65anim2d 587 . . 3 (𝐴𝐵 → ((𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)) → (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵))))
7 islly 21081 . . 3 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
8 islly 21081 . . 3 (𝑗 ∈ Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
96, 7, 83imtr4g 284 . 2 (𝐴𝐵 → (𝑗 ∈ Locally 𝐴𝑗 ∈ Locally 𝐵))
109ssrdv 3574 1 (𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Locally clly 21077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lly 21079 This theorem is referenced by: (None)
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