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Theorem llyss 22853
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 3941 . . . . . . . 8 (𝐴𝐵 → ((𝑗t 𝑢) ∈ 𝐴 → (𝑗t 𝑢) ∈ 𝐵))
21anim2d 613 . . . . . . 7 (𝐴𝐵 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
32reximdv 3164 . . . . . 6 (𝐴𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
43ralimdv 3163 . . . . 5 (𝐴𝐵 → (∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
54ralimdv 3163 . . . 4 (𝐴𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
65anim2d 613 . . 3 (𝐴𝐵 → ((𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)) → (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵))))
7 islly 22842 . . 3 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
8 islly 22842 . . 3 (𝑗 ∈ Locally 𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐵)))
96, 7, 83imtr4g 296 . 2 (𝐴𝐵 → (𝑗 ∈ Locally 𝐴𝑗 ∈ Locally 𝐵))
109ssrdv 3954 1 (𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  wrex 3070  cin 3913  wss 3914  𝒫 cpw 4564  (class class class)co 7361  t crest 17310  Topctop 22265  Locally clly 22838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-lly 22840
This theorem is referenced by: (None)
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