metustrel - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > metustrel		Structured version Visualization version GIF version

Theorem metustrel 22167

Description: Elements of the filter base generated by the metric 𝐷 are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**Hypothesis**
Ref	Expression
metust.1	⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))

**Assertion**
Ref	Expression
metustrel	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel 𝐴)

Distinct variable groups: 𝐷,𝑎 𝑋,𝑎 𝐴,𝑎 𝐹,𝑎

**Proof of Theorem metustrel**
Step	Hyp	Ref	Expression
1		metust.1	. . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
2	1	metustss 22166	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3		xpss 5149	. . 3 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
4	2, 3	syl6ss 3580	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (V × V))
5		df-rel 5045	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
6	4, 5	sylibr 223	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ran crn 5039 “ cima 5041 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℝ⁺crp 11708 [,)cico 12048 PsMetcpsmet 19551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-xr 9957 df-psmet 19559

This theorem is referenced by: (None)

W3C validator