Theorem ply1frcl 22238

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 4300	. . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅)
2		ply1frcl.q	. . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅)
3	1, 2	eleq2s 2846	. 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅)
4		rneq 5889	. . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅)
5		rn0 5879	. . . 4 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2780	. . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅)
7	6	necon3i 2957	. 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅)
8		n0 4312	. . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅))
9		df-evls1 22235	. . . . . . 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 8024	. . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))
11		elfvdm 6877	. . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
12		df-ov 7372	. . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩)
13	11, 12	eleq2s 2846	. . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 )
14	10, 13	sselid 3941	. . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)))
15		fveq2 6840	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
16	15	pweqd 4576	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆))
17	16	opeliunxp2 5792	. . . . 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 1930	. . 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  Vcvv 3444  ⦋csb 3859  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ⟨cop 4591  ∪ ciun 4951   ↦ cmpt 5183   × cxp 5629  dom cdm 5631  ran crn 5632   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369  1_oc1o 8404   ↑_m cmap 8776  Basecbs 17155   evalSub ces 22012   evalSub₁ ces1 22233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-evls1 22235

This theorem is referenced by: (None)