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Theorem stdbdmetval 18544
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 6106 . . . 4  |-  ( A C B )  e. 
_V
2 ifexg 3798 . . . 4  |-  ( ( ( A C B )  e.  _V  /\  R  e.  V )  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
31, 2mpan 652 . . 3  |-  ( R  e.  V  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
4 oveq12 6090 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x C y )  =  ( A C B ) )
54breq1d 4222 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x C y )  <_  R  <->  ( A C B )  <_  R ) )
6 eqidd 2437 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  R )
75, 4, 6ifbieq12d 3761 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
) )
8 stdbdmet.1 . . . 4  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
97, 8ovmpt2ga 6203 . . 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 ) )
103, 9syl3an3 1219 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  R  e.  V )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
11103comr 1161 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   ifcif 3739   class class class wbr 4212  (class class class)co 6081    e. cmpt2 6083    <_ cle 9121
This theorem is referenced by:  stdbdbl  18547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086
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