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

Theorem reldmdsmm 18891
Description: The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Assertion
Ref Expression
reldmdsmm  |-  Rel  dom  (+)m

Proof of Theorem reldmdsmm
Dummy variables  s 
r  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dsmm 18890 . 2  |-  (+)m  =  ( s  e.  _V , 
r  e.  _V  |->  ( ( s X_s r )s  { f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin } ) )
21reldmmpt2 6412 1  |-  Rel  dom  (+)m
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   dom cdm 5008   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   Basecbs 14644   ↾s cress 14645   0gc0g 14857   X_scprds 14863    (+)m cdsmm 18889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-dm 5018  df-oprab 6300  df-mpt2 6301  df-dsmm 18890
This theorem is referenced by:  dsmmval  18892  dsmmval2  18894
  Copyright terms: Public domain W3C validator