Theorem reldmdsmm 21952

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 21951	. 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 7551	1 ⊢ Rel dom ⊕m

This proof depends on syntax axioms:   ∈ wcel 2145   ≠ wne 2957  {crab 3414  Vcvv 3453  dom cdm 5659  Rel wrel 5664  ‘cfv 6537  (class class class)co 7417  Xcixp 8908  Fincfn 8956  Basecbs 17307   ↾_s cress 17328  0_gc0g 17530  X_scprds 17536   ⊕_m cdsmm 21950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669  df-oprab 7421  df-mpo 7422  df-dsmm 21951

This theorem is used by:  dsmmval  21953  dsmmval2  21955