ballss3 - Mathbox for Glauco Siliprandi

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Theorem ballss3 41729

Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
ballss3.y	⊢ Ⅎ𝑥𝜑
ballss3.d	⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))
ballss3.p	⊢ (𝜑 → 𝑃 ∈ 𝑋)
ballss3.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
ballss3.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)

**Assertion**
Ref	Expression
ballss3	⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐷 𝑥,𝑃 𝑥,𝑅 Allowed substitution hints: 𝜑(𝑥) 𝑋(𝑥)

**Proof of Theorem ballss3**
Step	Hyp	Ref	Expression
1		ballss3.y	. . 3 ⊢ Ⅎ𝑥𝜑
2		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
4		ballss3.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))
5		ballss3.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
6		ballss3.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		elblps 22994	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
8	4, 5, 6, 7	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
9	8	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
10	3, 9	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
11	10	simpld 498	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
12	10	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13		ballss3.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)
14	2, 11, 12, 13	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴)
15	14	ex 416	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴))
16	1, 15	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
17		dfss3 3903	. 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
18	16, 17	sylibr 237	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝ^*cxr 10663 < clt 10664 PsMetcpsmet 20075 ballcbl 20078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-xr 10668 df-psmet 20083 df-bl 20086

This theorem is referenced by: ioorrnopnlem 42946

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