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Theorem metidss 28574
Description: As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidss  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )

Proof of Theorem metidss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidval 28573 . 2  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  =  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) } )
2 opabssxp 4920 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  C_  ( X  X.  X
)
31, 2syl6eqss 3511 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    C_ wss 3433   {copab 4474    X. cxp 4843   ` cfv 5592  (class class class)co 6296   0cc0 9528  PsMetcpsmet 18895  ~Metcmetid 28569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7473  df-xr 9668  df-psmet 18903  df-metid 28571
This theorem is referenced by:  metideq  28576  metider  28577  pstmfval  28579
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