Theorem finxp1o 37380

Description: The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)

Assertion

Ref	Expression
finxp1o	⊢ (𝑈↑↑1o) = 𝑈

Proof of Theorem finxp1o

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1onn 8604	. . . . . 6 ⊢ 1o ∈ ω
2	1	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → 1o ∈ ω)
3		finxpreclem1 37377	. . . . . 6 ⊢ (𝑦 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
4		1on 8446	. . . . . . . 8 ⊢ 1o ∈ On
5		1n0 8452	. . . . . . . 8 ⊢ 1o ≠ ∅
6		nnlim 7856	. . . . . . . . 9 ⊢ (1o ∈ ω → ¬ Lim 1o)
7	1, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ Lim 1o
8		rdgsucuni 37357	. . . . . . . 8 ⊢ ((1o ∈ On ∧ 1o ≠ ∅ ∧ ¬ Lim 1o) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∪ 1o)))
9	4, 5, 7, 8	mp3an 1463	. . . . . . 7 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∪ 1o))
10		df-1o 8434	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
11	10	unieqi 4883	. . . . . . . . . . 11 ⊢ ∪ 1o = ∪ suc ∅
12		0elon 6387	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13	12	onunisuci 6454	. . . . . . . . . . 11 ⊢ ∪ suc ∅ = ∅
14	11, 13	eqtri 2752	. . . . . . . . . 10 ⊢ ∪ 1o = ∅
15	14	fveq2i 6861	. . . . . . . . 9 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∪ 1o) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∅)
16		opex 5424	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
17	16	rdg0 8389	. . . . . . . . 9 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∅) = ⟨1o, 𝑦⟩
18	15, 17	eqtri 2752	. . . . . . . 8 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∪ 1o) = ⟨1o, 𝑦⟩
19	18	fveq2i 6861	. . . . . . 7 ⊢ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∪ 1o)) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩)
20	9, 19	eqtri 2752	. . . . . 6 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩)
21	3, 20	eqtr4di 2782	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
22		df-finxp 37372	. . . . . 6 ⊢ (𝑈↑↑1o) = {𝑦 ∣ (1o ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))}
23	22	eqabri 2871	. . . . 5 ⊢ (𝑦 ∈ (𝑈↑↑1o) ↔ (1o ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o)))
24	2, 21, 23	sylanbrc 583	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (𝑈↑↑1o))
25	1, 23	mpbiran 709	. . . . 5 ⊢ (𝑦 ∈ (𝑈↑↑1o) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
26		vex 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
27	20	eqcomi 2738	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o)
28		finxpreclem2 37378	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
29	28	neqned 2932	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
30	29	necomd 2980	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩) ≠ ∅)
31	27, 30	eqnetrrid 3000	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) ≠ ∅)
32	31	necomd 2980	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → ∅ ≠ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
33	32	neneqd 2930	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑈) → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
34	26, 33	mpan 690	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑈 → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
35	34	con4i 114	. . . . 5 ⊢ (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) → 𝑦 ∈ 𝑈)
36	25, 35	sylbi 217	. . . 4 ⊢ (𝑦 ∈ (𝑈↑↑1o) → 𝑦 ∈ 𝑈)
37	24, 36	impbii 209	. . 3 ⊢ (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ (𝑈↑↑1o))
38	37	eqriv 2726	. 2 ⊢ 𝑈 = (𝑈↑↑1o)
39	38	eqcomi 2738	1 ⊢ (𝑈↑↑1o) = 𝑈

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ∅c0 4296  ifcif 4488  ⟨cop 4595  ∪ cuni 4871   × cxp 5636  Oncon0 6332  Lim wlim 6333  suc csuc 6334  ‘cfv 6511   ∈ cmpo 7389  ωcom 7842  1^st c1st 7966  reccrdg 8377  1_oc1o 8427  ↑↑cfinxp 37371

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-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-finxp 37372

This theorem is referenced by:  finxp2o  37387  finxp00  37390