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Theorem reldmdsmm 21155
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 21154 . 2 m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))
21reldmmpo 7491 1 Rel dom ⊕m
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wne 2940  {crab 3406  Vcvv 3444  dom cdm 5634  Rel wrel 5639  cfv 6497  (class class class)co 7358  Xcixp 8838  Fincfn 8886  Basecbs 17088  s cress 17117  0gc0g 17326  Xscprds 17332  m cdsmm 21153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-dm 5644  df-oprab 7362  df-mpo 7363  df-dsmm 21154
This theorem is referenced by:  dsmmval  21156  dsmmval2  21158
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