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Theorem reldmdsmm 19373
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 19372 . 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 6426 1  |-  Rel  dom  (+)m
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   dom cdm 4839   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   X_cixp 7540   Fincfn 7587   Basecbs 15199   ↾s cress 15200   0gc0g 15416   X_scprds 15422    (+)m cdsmm 19371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-dm 4849  df-oprab 6312  df-mpt2 6313  df-dsmm 19372
This theorem is referenced by:  dsmmval  19374  dsmmval2  19376
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