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Theorem psmetdmdm 22915

Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)

**Assertion**
Ref	Expression
psmetdmdm	⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)

Proof of Theorem psmetdmdm Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6703	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2		ispsmet 22914	. . . . . 6 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
3	2	biimpa 479	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
4	1, 3	mpancom 686	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
5	4	simpld 497	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
6		fdm 6522	. . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom 𝐷 = (𝑋 × 𝑋))
7	6	dmeqd 5774	. . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom dom 𝐷 = dom (𝑋 × 𝑋))
8	5, 7	syl 17	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
9		dmxpid 5800	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	syl6req 2873	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 class class class wbr 5066 × cxp 5553 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℝ^*cxr 10674 ≤ cle 10676 +_𝑒 cxad 12506 PsMetcpsmet 20529

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-xr 10679 df-psmet 20537

This theorem is referenced by: blfvalps 22993 metuval 23159 metidval 31130 pstmval 31135

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