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Theorem llyss 22658
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 3916 . . . . . . . 8 (𝐴𝐵 → ((𝑗t 𝑢) ∈ 𝐴 → (𝑗t 𝑢) ∈ 𝐵))
21anim2d 611 . . . . . . 7 (𝐴𝐵 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
32reximdv 3161 . . . . . 6 (𝐴𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
43ralimdv 3160 . . . . 5 (𝐴𝐵 → (∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
54ralimdv 3160 . . . 4 (𝐴𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
65anim2d 611 . . 3 (𝐴𝐵 → ((𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)) → (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵))))
7 islly 22647 . . 3 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
8 islly 22647 . . 3 (𝑗 ∈ Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
96, 7, 83imtr4g 295 . 2 (𝐴𝐵 → (𝑗 ∈ Locally 𝐴𝑗 ∈ Locally 𝐵))
109ssrdv 3929 1 (𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2101  wral 3059  wrex 3068  cin 3888  wss 3889  𝒫 cpw 4536  (class class class)co 7295  t crest 17159  Topctop 22070  Locally clly 22643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-lly 22645
This theorem is referenced by: (None)
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