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Theorem finxp1o 33511
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.
StepHypRef Expression
1 1onn 7876 . . . . . 6 1𝑜 ∈ ω
21a1i 11 . . . . 5 (𝑦𝑈 → 1𝑜 ∈ ω)
3 finxpreclem1 33508 . . . . . 6 (𝑦𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
4 1on 7724 . . . . . . . 8 1𝑜 ∈ On
5 1n0 7732 . . . . . . . 8 1𝑜 ≠ ∅
6 nnlim 7231 . . . . . . . . 9 (1𝑜 ∈ ω → ¬ Lim 1𝑜)
71, 6ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
8 rdgsucuni 33499 . . . . . . . 8 ((1𝑜 ∈ On ∧ 1𝑜 ≠ ∅ ∧ ¬ Lim 1𝑜) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜)))
94, 5, 7, 8mp3an 1561 . . . . . . 7 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜))
10 df-1o 7717 . . . . . . . . . . . 12 1𝑜 = suc ∅
1110unieqi 4585 . . . . . . . . . . 11 1𝑜 = suc ∅
12 0elon 5927 . . . . . . . . . . . 12 ∅ ∈ On
1312onunisuci 5990 . . . . . . . . . . 11 suc ∅ = ∅
1411, 13eqtri 2770 . . . . . . . . . 10 1𝑜 = ∅
1514fveq2i 6343 . . . . . . . . 9 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘∅)
16 opex 5069 . . . . . . . . . 10 ⟨1𝑜, 𝑦⟩ ∈ V
1716rdg0 7674 . . . . . . . . 9 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘∅) = ⟨1𝑜, 𝑦
1815, 17eqtri 2770 . . . . . . . 8 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜) = ⟨1𝑜, 𝑦
1918fveq2i 6343 . . . . . . 7 ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘(rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜)) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩)
209, 19eqtri 2770 . . . . . 6 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩)
213, 20syl6eqr 2800 . . . . 5 (𝑦𝑈 → ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
22 df-finxp 33503 . . . . . 6 (𝑈↑↑1𝑜) = {𝑦 ∣ (1𝑜 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))}
2322abeq2i 2861 . . . . 5 (𝑦 ∈ (𝑈↑↑1𝑜) ↔ (1𝑜 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜)))
242, 21, 23sylanbrc 701 . . . 4 (𝑦𝑈𝑦 ∈ (𝑈↑↑1𝑜))
251, 23mpbiran 991 . . . . 5 (𝑦 ∈ (𝑈↑↑1𝑜) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
26 vex 3331 . . . . . . 7 𝑦 ∈ V
2720eqcomi 2757 . . . . . . . . . 10 ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜)
28 finxpreclem2 33509 . . . . . . . . . . . 12 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
2928neqned 2927 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
3029necomd 2975 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩) ≠ ∅)
3127, 30syl5eqner 2995 . . . . . . . . 9 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) ≠ ∅)
3231necomd 2975 . . . . . . . 8 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3332neneqd 2925 . . . . . . 7 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3426, 33mpan 708 . . . . . 6 𝑦𝑈 → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3534con4i 113 . . . . 5 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) → 𝑦𝑈)
3625, 35sylbi 207 . . . 4 (𝑦 ∈ (𝑈↑↑1𝑜) → 𝑦𝑈)
3724, 36impbii 199 . . 3 (𝑦𝑈𝑦 ∈ (𝑈↑↑1𝑜))
3837eqriv 2745 . 2 𝑈 = (𝑈↑↑1𝑜)
3938eqcomi 2757 1 (𝑈↑↑1𝑜) = 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1620  wcel 2127  wne 2920  Vcvv 3328  c0 4046  ifcif 4218  cop 4315   cuni 4576   × cxp 5252  Oncon0 5872  Lim wlim 5873  suc csuc 5874  cfv 6037  cmpt2 6803  ωcom 7218  1st c1st 7319  reccrdg 7662  1𝑜c1o 7710  ↑↑cfinxp 33502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-finxp 33503
This theorem is referenced by:  finxp2o  33518  finxp00  33521
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