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Theorem metreslem 18384
Description: Lemma for metres 18387. (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 5353 . 2  |-  ( D  |`  dom  ( D  |`  ( R  X.  R
) ) )  =  ( D  |`  ( R  X.  R ) )
2 ineq2 3528 . . . 4  |-  ( dom 
D  =  ( X  X.  X )  -> 
( ( R  X.  R )  i^i  dom  D )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) ) )
3 dmres 5159 . . . 4  |-  dom  ( D  |`  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  dom  D )
4 inxp 4999 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R ) )
5 incom 3525 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) )
64, 5eqtr3i 2457 . . . 4  |-  ( ( X  i^i  R )  X.  ( X  i^i  R ) )  =  ( ( R  X.  R
)  i^i  ( X  X.  X ) )
72, 3, 63eqtr4g 2492 . . 3  |-  ( dom 
D  =  ( X  X.  X )  ->  dom  ( D  |`  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )
87reseq2d 5138 . 2  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  dom  ( D  |`  ( R  X.  R ) ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
91, 8syl5eqr 2481 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 1652    i^i cin 3311    X. cxp 4868   dom cdm 4870    |` cres 4872
This theorem is referenced by:  xmetres  18386  metres  18387
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882
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