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Theorem reldmdsmm 20017
 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 20016 . 2 m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))
21reldmmpt2 6736 1 Rel dom ⊕m
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1987   ≠ wne 2790  {crab 2912  Vcvv 3190  dom cdm 5084  Rel wrel 5089  ‘cfv 5857  (class class class)co 6615  Xcixp 7868  Fincfn 7915  Basecbs 15800   ↾s cress 15801  0gc0g 16040  Xscprds 16046   ⊕m cdsmm 20015 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-dm 5094  df-oprab 6619  df-mpt2 6620  df-dsmm 20016 This theorem is referenced by:  dsmmval  20018  dsmmval2  20020
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