Theorem reldmmpl 21980

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 21905	. 2 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
2	1	reldmmpo 7496	1 ⊢ Rel dom mPoly

Syntax hints:  {crab 3390  Vcvv 3430  ⦋csb 3838   class class class wbr 5086  dom cdm 5626  Rel wrel 5631  ‘cfv 6494  (class class class)co 7362   finSupp cfsupp 9269  Basecbs 17174   ↾_s cress 17195  0_gc0g 17397   mPwSer cmps 21898   mPoly cmpl 21900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5632  df-rel 5633  df-dm 5636  df-oprab 7366  df-mpo 7367  df-mpl 21905

This theorem is referenced by:  mplval  21981  mplrcl  21986  selvval  22115  ismhp  22120  psdmplcl  22142  mplbaspropd  22214  ply1ascl  22237  mdegfval  26041