Theorem finxp1o 37641

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 8578	. . . . . 6 ⊢ 1o ∈ ω
2	1	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → 1o ∈ ω)
3		finxpreclem1 37638	. . . . . 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 8419	. . . . . . . 8 ⊢ 1o ∈ On
5		1n0 8425	. . . . . . . 8 ⊢ 1o ≠ ∅
6		nnlim 7832	. . . . . . . . 9 ⊢ (1o ∈ ω → ¬ Lim 1o)
7	1, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ Lim 1o
8		rdgsucuni 37618	. . . . . . . 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 1464	. . . . . . 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 8407	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
11	10	unieqi 4877	. . . . . . . . . . 11 ⊢ ∪ 1o = ∪ suc ∅
12		0elon 6380	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13	12	onunisuci 6446	. . . . . . . . . . 11 ⊢ ∪ suc ∅ = ∅
14	11, 13	eqtri 2760	. . . . . . . . . 10 ⊢ ∪ 1o = ∅
15	14	fveq2i 6845	. . . . . . . . 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 5419	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
17	16	rdg0 8362	. . . . . . . . 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 6845	. . . . . . 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 37633	. . . . . 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 2879	. . . . 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 584	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (𝑈↑↑1o))
25	1, 23	mpbiran 710	. . . . 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 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
27	20	eqcomi 2746	. . . . . . . . . 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 37639	. . . . . . . . . . . 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 2940	. . . . . . . . . . 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 2988	. . . . . . . . . 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 3008	. . . . . . . . 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 2988	. . . . . . . 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 2938	. . . . . . 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 691	. . . . . 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 2734	. 2 ⊢ 𝑈 = (𝑈↑↑1o)
39	38	eqcomi 2746	1 ⊢ (𝑈↑↑1o) = 𝑈

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  ifcif 4481  ⟨cop 4588  ∪ cuni 4865   × cxp 5630  Oncon0 6325  Lim wlim 6326  suc csuc 6327  ‘cfv 6500   ∈ cmpo 7370  ωcom 7818  1^st c1st 7941  reccrdg 8350  1_oc1o 8400  ↑↑cfinxp 37632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-finxp 37633

This theorem is referenced by:  finxp2o  37648  finxp00  37651