Theorem reldmdsmm 22045

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 22044	. 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 7555	1 ⊢ Rel dom ⊕m

This proof depends on syntax axioms:   ∈ wcel 2144   ≠ wne 2955  {crab 3412  Vcvv 3450  dom cdm 5650  Rel wrel 5655  ‘cfv 6539  (class class class)co 7421  Xcixp 8929  Fincfn 8977  Basecbs 17393   ↾_s cress 17414  0_gc0g 17616  X_scprds 17622   ⊕_m cdsmm 22043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-oprab 7425  df-mpo 7426  df-dsmm 22044

This theorem is used by:  dsmmval  22046  dsmmval2  22048