Theorem ply1frcl 22270

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 4321	. . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2		ply1frcl.q	. . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅)
3	1, 2	eleq2s 2851	. 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4		rneq 5927	. . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5		rn0 5916	. . . 4 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2785	. . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
7	6	necon3i 2963	. 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8		n0 4333	. . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9		df-evls1 22267	. . . . . . 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 8073	. . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11		elfvdm 6923	. . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12		df-ov 7416	. . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
13	11, 12	eleq2s 2851	. . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
14	10, 13	sselid 3961	. . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15		fveq2 6886	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
16	15	pweqd 4597	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
17	16	opeliunxp2 5829	. . . . 5 ⊢ (⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
18	14, 17	sylib 218	. . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
19	18	exlimiv 1929	. . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
20	8, 19	sylbi 217	. 2 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
21	3, 7, 20	3syl 18	1 ⊢ (𝑋 ∈ 𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2931  Vcvv 3463  ⦋csb 3879  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ⟨cop 4612  ∪ ciun 4971   ↦ cmpt 5205   × cxp 5663  dom cdm 5665  ran crn 5666   ∘ ccom 5669  ‘cfv 6541  (class class class)co 7413  1_oc1o 8481   ↑_m cmap 8848  Basecbs 17229   evalSub ces 22044   evalSub₁ ces1 22265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-evls1 22267

This theorem is referenced by: (None)