Theorem finxp1o 34667

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 8259	. . . . . 6 ⊢ 1o ∈ ω
2	1	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → 1o ∈ ω)
3		finxpreclem1 34664	. . . . . 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 8103	. . . . . . . 8 ⊢ 1o ∈ On
5		1n0 8113	. . . . . . . 8 ⊢ 1o ≠ ∅
6		nnlim 7587	. . . . . . . . 9 ⊢ (1o ∈ ω → ¬ Lim 1o)
7	1, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ Lim 1o
8		rdgsucuni 34644	. . . . . . . 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 1457	. . . . . . 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 8096	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
11	10	unieqi 4840	. . . . . . . . . . 11 ⊢ ∪ 1o = ∪ suc ∅
12		0elon 6238	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13	12	onunisuci 6298	. . . . . . . . . . 11 ⊢ ∪ suc ∅ = ∅
14	11, 13	eqtri 2844	. . . . . . . . . 10 ⊢ ∪ 1o = ∅
15	14	fveq2i 6667	. . . . . . . . 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 5348	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
17	16	rdg0 8051	. . . . . . . . 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 2844	. . . . . . . 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 6667	. . . . . . 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 2844	. . . . . 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	syl6eqr 2874	. . . . 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 34659	. . . . . 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	abeq2i 2948	. . . . 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 585	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (𝑈↑↑1o))
25	1, 23	mpbiran 707	. . . . 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 3497	. . . . . . 7 ⊢ 𝑦 ∈ V
27	20	eqcomi 2830	. . . . . . . . . 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 34665	. . . . . . . . . . . 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 3023	. . . . . . . . . . 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 3071	. . . . . . . . . 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 3091	. . . . . . . . 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 3071	. . . . . . . 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 3021	. . . . . . 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 688	. . . . . 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 219	. . . 4 ⊢ (𝑦 ∈ (𝑈↑↑1o) → 𝑦 ∈ 𝑈)
37	24, 36	impbii 211	. . 3 ⊢ (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ (𝑈↑↑1o))
38	37	eqriv 2818	. 2 ⊢ 𝑈 = (𝑈↑↑1o)
39	38	eqcomi 2830	1 ⊢ (𝑈↑↑1o) = 𝑈

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290  ifcif 4466  ⟨cop 4566  ∪ cuni 4831   × cxp 5547  Oncon0 6185  Lim wlim 6186  suc csuc 6187  ‘cfv 6349   ∈ cmpo 7152  ωcom 7574  1^st c1st 7681  reccrdg 8039  1_oc1o 8089  ↑↑cfinxp 34658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-finxp 34659

This theorem is referenced by:  finxp2o  34674  finxp00  34677