Theorem llyss 23705

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 3925	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 → (𝑗 ↾t 𝑢) ∈ 𝐵))
2	1	anim2d 624	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)))
3	2	reximdv 3177	. . . . . 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 3176	. . . . 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 3176	. . . 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 624	. . 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 23694	. . 3 ⊢ (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)))
8		islly 23694	. . 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 299	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Locally 𝐵))
10	9	ssrdv 3937	1 ⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  (class class class)co 7413   ↾_t crest 17505  Topctop 23118  Locally clly 23690

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-lly 23692

This theorem is used by: (None)