Theorem finxp1o 36359

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 8641	. . . . . 6 ⊢ 1o ∈ ω
2	1	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → 1o ∈ ω)
3		finxpreclem1 36356	. . . . . 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 8480	. . . . . . . 8 ⊢ 1o ∈ On
5		1n0 8490	. . . . . . . 8 ⊢ 1o ≠ ∅
6		nnlim 7871	. . . . . . . . 9 ⊢ (1o ∈ ω → ¬ Lim 1o)
7	1, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ Lim 1o
8		rdgsucuni 36336	. . . . . . . 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 1461	. . . . . . 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 8468	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
11	10	unieqi 4921	. . . . . . . . . . 11 ⊢ ∪ 1o = ∪ suc ∅
12		0elon 6418	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13	12	onunisuci 6484	. . . . . . . . . . 11 ⊢ ∪ suc ∅ = ∅
14	11, 13	eqtri 2760	. . . . . . . . . 10 ⊢ ∪ 1o = ∅
15	14	fveq2i 6894	. . . . . . . . 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 5464	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
17	16	rdg0 8423	. . . . . . . . 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 2760	. . . . . . . 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 6894	. . . . . . 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 2760	. . . . . 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 2790	. . . . 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 36351	. . . . . 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 2877	. . . . 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 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 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
27	20	eqcomi 2741	. . . . . . . . . 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 36357	. . . . . . . . . . . 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 2947	. . . . . . . . . . 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 2996	. . . . . . . . . 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 3016	. . . . . . . . 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 2996	. . . . . . . 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 2945	. . . . . . 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 216	. . . 4 ⊢ (𝑦 ∈ (𝑈↑↑1o) → 𝑦 ∈ 𝑈)
37	24, 36	impbii 208	. . 3 ⊢ (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ (𝑈↑↑1o))
38	37	eqriv 2729	. 2 ⊢ 𝑈 = (𝑈↑↑1o)
39	38	eqcomi 2741	1 ⊢ (𝑈↑↑1o) = 𝑈

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474  ∅c0 4322  ifcif 4528  ⟨cop 4634  ∪ cuni 4908   × cxp 5674  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ‘cfv 6543   ∈ cmpo 7413  ωcom 7857  1^st c1st 7975  reccrdg 8411  1_oc1o 8461  ↑↑cfinxp 36350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-finxp 36351

This theorem is referenced by:  finxp2o  36366  finxp00  36369