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Theorem ply1frcl 21700
Description: Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
Hypothesis
Ref Expression
ply1frcl.q 𝑄 = ran (𝑆 evalSub1 𝑅)
Assertion
Ref Expression
ply1frcl (𝑋𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))

Proof of Theorem ply1frcl
Dummy variables 𝑟 𝑏 𝑠 𝑥 𝑦 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 4299 . . 3 (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2 ply1frcl.q . . 3 𝑄 = ran (𝑆 evalSub1 𝑅)
31, 2eleq2s 2856 . 2 (𝑋𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4 rneq 5896 . . . 4 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5 rn0 5886 . . . 4 ran ∅ = ∅
64, 5eqtrdi 2793 . . 3 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
76necon3i 2977 . 2 (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8 n0 4311 . . 3 ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9 df-evls1 21697 . . . . . . 7 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
109dmmpossx 8003 . . . . . 6 dom evalSub1 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11 elfvdm 6884 . . . . . . 7 (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12 df-ov 7365 . . . . . . 7 (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
1311, 12eleq2s 2856 . . . . . 6 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
1410, 13sselid 3947 . . . . 5 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15 fveq2 6847 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1615pweqd 4582 . . . . . 6 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
1716opeliunxp2 5799 . . . . 5 (⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1814, 17sylib 217 . . . 4 (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1918exlimiv 1934 . . 3 (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
208, 19sylbi 216 . 2 ((𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
213, 7, 203syl 18 1 (𝑋𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  Vcvv 3448  csb 3860  c0 4287  𝒫 cpw 4565  {csn 4591  cop 4597   ciun 4959  cmpt 5193   × cxp 5636  dom cdm 5638  ran crn 5639  ccom 5642  cfv 6501  (class class class)co 7362  1oc1o 8410  m cmap 8772  Basecbs 17090   evalSub ces 21496   evalSub1 ces1 21695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-evls1 21697
This theorem is referenced by: (None)
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