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Theorem ply1frcl 20043
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 4150 . . 3 (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2 ply1frcl.q . . 3 𝑄 = ran (𝑆 evalSub1 𝑅)
31, 2eleq2s 2924 . 2 (𝑋𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4 rneq 5583 . . . 4 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5 rn0 5610 . . . 4 ran ∅ = ∅
64, 5syl6eq 2877 . . 3 ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
76necon3i 3031 . 2 (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8 n0 4160 . . 3 ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9 df-evls1 20040 . . . . . . 7 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
109dmmpt2ssx 7498 . . . . . 6 dom evalSub1 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11 elfvdm 6465 . . . . . . 7 (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12 df-ov 6908 . . . . . . 7 (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
1311, 12eleq2s 2924 . . . . . 6 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
1410, 13sseldi 3825 . . . . 5 (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15 fveq2 6433 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1615pweqd 4383 . . . . . 6 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
1716opeliunxp2 5493 . . . . 5 (⟨𝑆, 𝑅⟩ ∈ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1814, 17sylib 210 . . . 4 (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
1918exlimiv 2029 . . 3 (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
208, 19sylbi 209 . 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 386   = wceq 1656  wex 1878  wcel 2164  wne 2999  Vcvv 3414  csb 3757  c0 4144  𝒫 cpw 4378  {csn 4397  cop 4403   ciun 4740  cmpt 4952   × cxp 5340  dom cdm 5342  ran crn 5343  ccom 5346  cfv 6123  (class class class)co 6905  1oc1o 7819  𝑚 cmap 8122  Basecbs 16222   evalSub ces 19864   evalSub1 ces1 20038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-evls1 20040
This theorem is referenced by: (None)
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