Theorem reldmmpl 20209

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 20140	. 2 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
2	1	reldmmpo 7287	1 ⊢ Rel dom mPoly

Syntax hints:  {crab 3144  Vcvv 3496  ⦋csb 3885   class class class wbr 5068  dom cdm 5557  Rel wrel 5562  ‘cfv 6357  (class class class)co 7158   finSupp cfsupp 8835  Basecbs 16485   ↾_s cress 16486  0_gc0g 16715   mPwSer cmps 20133   mPoly cmpl 20135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-dm 5567  df-oprab 7162  df-mpo 7163  df-mpl 20140

This theorem is referenced by:  mplval  20210  mplrcl  20272  mplbaspropd  20407  ply1ascl  20428  mdegfval  24658  mdegcl  24665