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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 22354 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | 1 | fdmd 6193 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
3 | 2 | dmeqd 5463 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
4 | dmxpid 5482 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
5 | 3, 4 | syl6req 2822 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 × cxp 5248 dom cdm 5250 ‘cfv 6030 ℝ*cxr 10279 ∞Metcxmt 19946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-map 8015 df-xr 10284 df-xmet 19954 |
This theorem is referenced by: metdmdm 22361 xmetunirn 22362 cfilfval 23281 |
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