MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metreslem Unicode version

Theorem metreslem 18293
Description: Lemma for metres 18296. (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 5294 . 2  |-  ( D  |`  dom  ( D  |`  ( R  X.  R
) ) )  =  ( D  |`  ( R  X.  R ) )
2 ineq2 3472 . . . 4  |-  ( dom 
D  =  ( X  X.  X )  -> 
( ( R  X.  R )  i^i  dom  D )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) ) )
3 dmres 5100 . . . 4  |-  dom  ( D  |`  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  dom  D )
4 inxp 4940 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R ) )
5 incom 3469 . . . . 5  |-  ( ( X  X.  X )  i^i  ( R  X.  R ) )  =  ( ( R  X.  R )  i^i  ( X  X.  X ) )
64, 5eqtr3i 2402 . . . 4  |-  ( ( X  i^i  R )  X.  ( X  i^i  R ) )  =  ( ( R  X.  R
)  i^i  ( X  X.  X ) )
72, 3, 63eqtr4g 2437 . . 3  |-  ( dom 
D  =  ( X  X.  X )  ->  dom  ( D  |`  ( R  X.  R ) )  =  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )
87reseq2d 5079 . 2  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  dom  ( D  |`  ( R  X.  R ) ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
91, 8syl5eqr 2426 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 1649    i^i cin 3255    X. cxp 4809   dom cdm 4811    |` cres 4813
This theorem is referenced by:  xmetres  18295  metres  18296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823
  Copyright terms: Public domain W3C validator