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Theorem metreslem 13373
Description: Lemma for metres 13376. (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 5112 . 2  |-  ( D  |`  dom  ( D  |`  ( R  X.  R
) ) )  =  ( D  |`  ( R  X.  R ) )
2 ineq2 3328 . . . 4  |-  ( dom 
D  =  ( X  X.  X )  -> 
( ( R  X.  R )  i^i  dom  D )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) ) )
3 dmres 4921 . . . 4  |-  dom  ( D  |`  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  dom  D )
4 inxp 4754 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R ) )
5 incom 3325 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) )
64, 5eqtr3i 2198 . . . 4  |-  ( ( X  i^i  R )  X.  ( X  i^i  R ) )  =  ( ( R  X.  R
)  i^i  ( X  X.  X ) )
72, 3, 63eqtr4g 2233 . . 3  |-  ( dom 
D  =  ( X  X.  X )  ->  dom  ( D  |`  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )
87reseq2d 4900 . 2  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  dom  ( D  |`  ( R  X.  R ) ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
91, 8eqtr3id 2222 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 1353    i^i cin 3126    X. cxp 4618   dom cdm 4620    |` cres 4622
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632
This theorem is referenced by:  xmetres  13375  metres  13376
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