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Theorem ply1frcl 19956
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 4085 . . 3 (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2 ply1frcl.q . . 3 𝑄 = ran (𝑆 evalSub1 𝑅)
31, 2eleq2s 2862 . 2 (𝑋𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4 rneq 5519 . . . 4 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5 rn0 5546 . . . 4 ran ∅ = ∅
64, 5syl6eq 2815 . . 3 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
76necon3i 2969 . 2 (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8 n0 4095 . . 3 ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9 df-evls1 19953 . . . . . . 7 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
109dmmpt2ssx 7436 . . . . . 6 dom evalSub1 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11 elfvdm 6407 . . . . . . 7 (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12 df-ov 6845 . . . . . . 7 (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
1311, 12eleq2s 2862 . . . . . 6 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
1410, 13sseldi 3759 . . . . 5 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15 fveq2 6375 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1615pweqd 4320 . . . . . 6 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
1716opeliunxp2 5429 . . . . 5 (⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1814, 17sylib 209 . . . 4 (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1918exlimiv 2025 . . 3 (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
208, 19sylbi 208 . 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 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  Vcvv 3350  csb 3691  c0 4079  𝒫 cpw 4315  {csn 4334  cop 4340   ciun 4676  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278  ccom 5281  cfv 6068  (class class class)co 6842  1𝑜c1o 7757  𝑚 cmap 8060  Basecbs 16132   evalSub ces 19777   evalSub1 ces1 19951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-evls1 19953
This theorem is referenced by: (None)
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