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Theorem psmetdmdm 20991
Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
psmetdmdm  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )

Proof of Theorem psmetdmdm
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5830 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
2 ispsmet 20990 . . . . . 6  |-  ( X  e.  _V  ->  ( D  e.  (PsMet `  X
)  <->  ( D :
( X  X.  X
) --> RR*  /\  A. x  e.  X  ( (
x D x )  =  0  /\  A. y  e.  X  A. z  e.  X  (
x D y )  <_  ( ( z D x ) +e ( z D y ) ) ) ) ) )
32biimpa 482 . . . . 5  |-  ( ( X  e.  _V  /\  D  e.  (PsMet `  X
) )  ->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  (
( x D x )  =  0  /\ 
A. y  e.  X  A. z  e.  X  ( x D y )  <_  ( (
z D x ) +e ( z D y ) ) ) ) )
41, 3mpancom 667 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  (
( x D x )  =  0  /\ 
A. y  e.  X  A. z  e.  X  ( x D y )  <_  ( (
z D x ) +e ( z D y ) ) ) ) )
54simpld 457 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
6 fdm 5672 . . . 4  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
76dmeqd 5145 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
dom  D  =  dom  ( X  X.  X
) )
85, 7syl 17 . 2  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  dom  ( X  X.  X ) )
9 dmxpid 5162 . 2  |-  dom  ( X  X.  X )  =  X
108, 9syl6req 2458 1  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   _Vcvv 3056   class class class wbr 4392    X. cxp 4938   dom cdm 4940   -->wf 5519   ` cfv 5523  (class class class)co 6232   0cc0 9440   RR*cxr 9575    <_ cle 9577   +ecxad 11285  PsMetcpsmet 18612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-map 7377  df-xr 9580  df-psmet 18621
This theorem is referenced by:  blfvalps  21068  metuval  21235  metidval  28203  pstmval  28208
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