Theorem reldmdsmm 21020

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 21019	. 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 7449	1 ⊢ Rel dom ⊕m

Syntax hints:   ∈ wcel 2105   ≠ wne 2940  {crab 3403  Vcvv 3440  dom cdm 5607  Rel wrel 5612  ‘cfv 6465  (class class class)co 7316  Xcixp 8734  Fincfn 8782  Basecbs 16986   ↾_s cress 17015  0_gc0g 17224  X_scprds 17230   ⊕_m cdsmm 21018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5087  df-opab 5149  df-xp 5613  df-rel 5614  df-dm 5617  df-oprab 7320  df-mpo 7321  df-dsmm 21019

This theorem is referenced by:  dsmmval  21021  dsmmval2  21023