Theorem reldmdsmm 21713

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 21712	. 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 7501	1 ⊢ Rel dom ⊕m

Syntax hints:   ∈ wcel 2114   ≠ wne 2932  {crab 3389  Vcvv 3429  dom cdm 5631  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  Xcixp 8845  Fincfn 8893  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402  X_scprds 17408   ⊕_m cdsmm 21711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641  df-oprab 7371  df-mpo 7372  df-dsmm 21712

This theorem is referenced by:  dsmmval  21714  dsmmval2  21716