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Theorem metreslem 13540
Description: Lemma for metres 13543. (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 5116 . 2  |-  ( D  |`  dom  ( D  |`  ( R  X.  R
) ) )  =  ( D  |`  ( R  X.  R ) )
2 ineq2 3330 . . . 4  |-  ( dom 
D  =  ( X  X.  X )  -> 
( ( R  X.  R )  i^i  dom  D )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) ) )
3 dmres 4924 . . . 4  |-  dom  ( D  |`  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  dom  D )
4 inxp 4757 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R ) )
5 incom 3327 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) )
64, 5eqtr3i 2200 . . . 4  |-  ( ( X  i^i  R )  X.  ( X  i^i  R ) )  =  ( ( R  X.  R
)  i^i  ( X  X.  X ) )
72, 3, 63eqtr4g 2235 . . 3  |-  ( dom 
D  =  ( X  X.  X )  ->  dom  ( D  |`  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )
87reseq2d 4903 . 2  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  dom  ( D  |`  ( R  X.  R ) ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
91, 8eqtr3id 2224 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 3128    X. cxp 4621   dom cdm 4623    |` cres 4625
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635
This theorem is referenced by:  xmetres  13542  metres  13543
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