Theorem reldmmpl 22295

Description: The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
reldmmpl	⊢ Rel dom mPoly

Proof of Theorem reldmmpl

Dummy variables 𝑓 𝑖 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpl 22219	. 2 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
2	1	reldmmpo 7554	1 ⊢ Rel dom mPoly

This proof depends on syntax axioms:  {crab 3412  Vcvv 3450  ⦋csb 3846   class class class wbr 5102  dom cdm 5650  Rel wrel 5655  ‘cfv 6538  (class class class)co 7420   finSupp cfsupp 9353  Basecbs 17387   ↾_s cress 17408  0_gc0g 17610   mPwSer cmps 22212   mPoly cmpl 22214

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 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-oprab 7424  df-mpo 7425  df-mpl 22219

This theorem is used by:  mplval  22296  mplrcl  22301  selvval  22429  ismhp  22461  psdmplcl  22483  mplbaspropd  22554  ply1ascl  22577  mdegfval  26380