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Theorem psmetdmdm 20016
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 5829 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
2 ispsmet 20015 . . . . . 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 484 . . . . 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 669 . . . 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 459 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
6 fdm 5674 . . . 4  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
76dmeqd 5153 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
dom  D  =  dom  ( X  X.  X
) )
85, 7syl 16 . 2  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  dom  ( X  X.  X ) )
9 dmxpid 5170 . 2  |-  dom  ( X  X.  X )  =  X
108, 9syl6req 2512 1  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   class class class wbr 4403    X. cxp 4949   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203   0cc0 9396   RR*cxr 9531    <_ cle 9533   +ecxad 11201  PsMetcpsmet 17928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-xr 9536  df-psmet 17937
This theorem is referenced by:  blfvalps  20093  metuval  20260  metidval  26482  pstmval  26487
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