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Theorem metreslem 14357
Description: Lemma for metres 14360. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
metreslem  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )

Proof of Theorem metreslem
StepHypRef Expression
1 resdmres 5138 . 2  |-  ( D  |`  dom  ( D  |`  ( R  X.  R
) ) )  =  ( D  |`  ( R  X.  R ) )
2 ineq2 3345 . . . 4  |-  ( dom 
D  =  ( X  X.  X )  -> 
( ( R  X.  R )  i^i  dom  D )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) ) )
3 dmres 4946 . . . 4  |-  dom  ( D  |`  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  dom  D )
4 inxp 4779 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R ) )
5 incom 3342 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) )
64, 5eqtr3i 2212 . . . 4  |-  ( ( X  i^i  R )  X.  ( X  i^i  R ) )  =  ( ( R  X.  R
)  i^i  ( X  X.  X ) )
72, 3, 63eqtr4g 2247 . . 3  |-  ( dom 
D  =  ( X  X.  X )  ->  dom  ( D  |`  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )
87reseq2d 4925 . 2  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  dom  ( D  |`  ( R  X.  R ) ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
91, 8eqtr3id 2236 1  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    i^i cin 3143    X. cxp 4642   dom cdm 4644    |` cres 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656
This theorem is referenced by:  xmetres  14359  metres  14360
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