Theorem reldmmpl 15063

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 𝑎 𝑏 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mplcoe 15031	. 2 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))
2	1	reldmmpo 6194	1 ⊢ Rel dom mPoly

Syntax hints:   → wi 4   = wceq 1402  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821  ⦋csb 3147   class class class wbr 4128  dom cdm 4772  Rel wrel 4777  ‘cfv 5375  (class class class)co 6079   ↑_𝑚 cmap 6916   < clt 8354  ℕ₀cn0 9546  Basecbs 13335   ↾_s cress 13336  0_gc0g 13593   mPwSer cmps 15028   mPoly cmpl 15029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-dm 4782  df-oprab 6083  df-mpo 6084  df-mplcoe 15031

This theorem is referenced by:  mplrcl  15068  mplbasss  15070  mpladd  15078