Theorem reldmmpl 22031

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 21954	. 2 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
2	1	reldmmpo 7584	1 ⊢ Rel dom mPoly

Syntax hints:  {crab 3443  Vcvv 3488  ⦋csb 3921   class class class wbr 5166  dom cdm 5700  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448   finSupp cfsupp 9431  Basecbs 17258   ↾_s cress 17287  0_gc0g 17499   mPwSer cmps 21947   mPoly cmpl 21949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-oprab 7452  df-mpo 7453  df-mpl 21954

This theorem is referenced by:  mplval  22032  mplrcl  22037  mplbaspropd  22259  ply1ascl  22282  mdegfval  26121  mdegcl  26128