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| Mirrors > Home > MPE Home > Th. List > psmetdmdm | Structured version Visualization version GIF version | ||
| Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| Ref | Expression |
|---|---|
| psmetdmdm | β’ (π· β (PsMetβπ) β π = dom dom π·) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvex 6929 | . . . . 5 β’ (π· β (PsMetβπ) β π β V) | |
| 2 | ispsmet 24030 | . . . . . 6 β’ (π β V β (π· β (PsMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π ((π₯π·π₯) = 0 β§ βπ¦ β π βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))))) | |
| 3 | 2 | biimpa 477 | . . . . 5 β’ ((π β V β§ π· β (PsMetβπ)) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π ((π₯π·π₯) = 0 β§ βπ¦ β π βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))))) |
| 4 | 1, 3 | mpancom 686 | . . . 4 β’ (π· β (PsMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π ((π₯π·π₯) = 0 β§ βπ¦ β π βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))))) |
| 5 | 4 | simpld 495 | . . 3 β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆβ*) |
| 6 | fdm 6726 | . . . 4 β’ (π·:(π Γ π)βΆβ* β dom π· = (π Γ π)) | |
| 7 | 6 | dmeqd 5905 | . . 3 β’ (π·:(π Γ π)βΆβ* β dom dom π· = dom (π Γ π)) |
| 8 | 5, 7 | syl 17 | . 2 β’ (π· β (PsMetβπ) β dom dom π· = dom (π Γ π)) |
| 9 | dmxpid 5929 | . 2 β’ dom (π Γ π) = π | |
| 10 | 8, 9 | eqtr2di 2789 | 1 β’ (π· β (PsMetβπ) β π = dom dom π·) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 class class class wbr 5148 Γ cxp 5674 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7411 0cc0 11112 β*cxr 11251 β€ cle 11253 +π cxad 13094 PsMetcpsmet 21128 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 df-xr 11256 df-psmet 21136 |
| This theorem is referenced by: blfvalps 24109 metuval 24278 metidval 33156 pstmval 33161 |
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