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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 22941 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | 1 | fdmd 6525 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
3 | 2 | dmeqd 5776 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
4 | dmxpid 5802 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
5 | 3, 4 | syl6req 2875 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 × cxp 5555 dom cdm 5557 ‘cfv 6357 ℝ*cxr 10676 ∞Metcxmet 20532 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-xr 10681 df-xmet 20540 |
This theorem is referenced by: metdmdm 22948 xmetunirn 22949 cfilfval 23869 |
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