elblps - Metamath Proof Explorer

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Theorem elblps 24595

Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)

**Assertion**
Ref	Expression
elblps	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))

Proof of Theorem elblps Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		blvalps 24593	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
2	1	eleq2d 2851	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}))
3		oveq2 7427	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑃𝐷𝑥) = (𝑃𝐷𝐴))
4	3	breq1d 5121	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑃𝐷𝑥) < 𝑅 ↔ (𝑃𝐷𝐴) < 𝑅))
5	4	elrab 3652	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))
6	2, 5	bitrdi 290	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 {crab 3418 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 ℝ^*cxr 11257 < clt 11258 PsMetcpsmet 21556 ballcbl 21559

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172

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-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-map 8832 df-xr 11262 df-psmet 21564 df-bl 21567

This theorem is used by: elbl2ps 24597 xblpnfps 24603 xblss2ps 24609 xblcntrps 24618 blssps 24632 ballss3 45869

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