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Theorem stdbdmetval 21527
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
Assertion
Ref Expression
stdbdmetval  |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
Distinct variable groups:    x, y, A    x, C, y    x, B, y    x, R, y   
x, X, y
Allowed substitution hints:    D( x, y)    V( x, y)

Proof of Theorem stdbdmetval
StepHypRef Expression
1 ovex 6333 . . . 4  |-  ( A C B )  e. 
_V
2 ifexg 3980 . . . 4  |-  ( ( ( A C B )  e.  _V  /\  R  e.  V )  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
31, 2mpan 674 . . 3  |-  ( R  e.  V  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
4 oveq12 6314 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x C y )  =  ( A C B ) )
54breq1d 4433 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x C y )  <_  R  <->  ( A C B )  <_  R ) )
65, 4ifbieq1d 3934 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
) )
7 stdbdmet.1 . . . 4  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
86, 7ovmpt2ga 6440 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
93, 8syl3an3 1299 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  R  e.  V )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
1093comr 1213 1  |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3080   ifcif 3911   class class class wbr 4423  (class class class)co 6305    |-> cmpt2 6307    <_ cle 9683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310
This theorem is referenced by:  stdbdbl  21530
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