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Mirrors > Home > MPE Home > Th. List > xmetdmdm | Structured version Visualization version GIF version |
Description: Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xmetdmdm | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 23480 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | 1 | fdmd 6609 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
3 | 2 | dmeqd 5813 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
4 | dmxpid 5838 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
5 | 3, 4 | eqtr2di 2797 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 × cxp 5588 dom cdm 5590 ‘cfv 6432 ℝ*cxr 11009 ∞Metcxmet 20580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-map 8600 df-xr 11014 df-xmet 20588 |
This theorem is referenced by: metdmdm 23487 xmetunirn 23488 cfilfval 24426 |
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