Theorem reldmdsmm 20940

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.

Step	Hyp	Ref	Expression
1		df-dsmm 20939	. 2 ⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}))
2	1	reldmmpo 7408	1 ⊢ Rel dom ⊕m

Syntax hints:   ∈ wcel 2106   ≠ wne 2943  {crab 3068  Vcvv 3432  dom cdm 5589  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275  Xcixp 8685  Fincfn 8733  Basecbs 16912   ↾_s cress 16941  0_gc0g 17150  X_scprds 17156   ⊕_m cdsmm 20938

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-oprab 7279  df-mpo 7280  df-dsmm 20939

This theorem is referenced by:  dsmmval  20941  dsmmval2  20943