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Theorem ply1frcl 19504
 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 3880 . . 3 (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2 ply1frcl.q . . 3 𝑄 = ran (𝑆 evalSub1 𝑅)
31, 2eleq2s 2706 . 2 (𝑋𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4 rneq 5272 . . . 4 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5 rn0 5298 . . . 4 ran ∅ = ∅
64, 5syl6eq 2660 . . 3 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
76necon3i 2814 . 2 (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8 n0 3890 . . 3 ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9 df-evls1 19501 . . . . . . 7 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
109dmmpt2ssx 7124 . . . . . 6 dom evalSub1 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11 elfvdm 6130 . . . . . . 7 (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12 df-ov 6552 . . . . . . 7 (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
1311, 12eleq2s 2706 . . . . . 6 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
1410, 13sseldi 3566 . . . . 5 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15 fveq2 6103 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1615pweqd 4113 . . . . . 6 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
1716opeliunxp2 5182 . . . . 5 (⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1814, 17sylib 207 . . . 4 (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1918exlimiv 1845 . . 3 (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
208, 19sylbi 206 . 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 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ⦋csb 3499  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744  Basecbs 15695   evalSub ces 19325   evalSub1 ces1 19499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-evls1 19501 This theorem is referenced by: (None)
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