Theorem ply1frcl 22489

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	⊢ (𝑋 ∈ 𝑄 → (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.

Step	Hyp	Ref	Expression
1		ne0i 4293	. . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2		ply1frcl.q	. . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅)
3	1, 2	eleq2s 2880	. 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4		rneq 5925	. . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5		rn0 5915	. . . 4 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2813	. . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
7	6	necon3i 2989	. 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8		n0 4306	. . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9		df-evls1 22486	. . . . . . 7 ⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
10	9	dmmpossx 8061	. . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11		elfvdm 6915	. . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12		df-ov 7415	. . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
13	11, 12	eleq2s 2880	. . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
14	10, 13	sselid 3934	. . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15		fveq2 6881	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
16	15	pweqd 4578	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
17	16	opeliunxp2 5823	. . . . 5 ⊢ (⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
18	14, 17	sylib 221	. . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
19	18	exlimiv 1959	. . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
20	8, 19	sylbi 220	. 2 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
21	3, 7, 20	3syl 19	1 ⊢ (𝑋 ∈ 𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454  ⦋csb 3852  ∅c0 4285  𝒫 cpw 4561  {csn 4588  ⟨cop 4594  ∪ ciun 4955   ↦ cmpt 5191   × cxp 5658  dom cdm 5660  ran crn 5661   ∘ ccom 5664  ‘cfv 6536  (class class class)co 7412  1_oc1o 8444   ↑_m cmap 8822  Basecbs 17275   evalSub ces 22234   evalSub₁ ces1 22484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-evls1 22486

This theorem is used by: (None)