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Theorem finxp1o 32858
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 7664 . . . . . 6 1𝑜 ∈ ω
21a1i 11 . . . . 5 (𝑦𝑈 → 1𝑜 ∈ ω)
3 finxpreclem1 32855 . . . . . 6 (𝑦𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
4 1on 7512 . . . . . . . 8 1𝑜 ∈ On
5 1n0 7520 . . . . . . . 8 1𝑜 ≠ ∅
6 nnlim 7025 . . . . . . . . 9 (1𝑜 ∈ ω → ¬ Lim 1𝑜)
71, 6ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
8 rdgsucuni 32846 . . . . . . . 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 1421 . . . . . . 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 7505 . . . . . . . . . . . 12 1𝑜 = suc ∅
1110unieqi 4411 . . . . . . . . . . 11 1𝑜 = suc ∅
12 0elon 5737 . . . . . . . . . . . 12 ∅ ∈ On
1312onunisuci 5800 . . . . . . . . . . 11 suc ∅ = ∅
1411, 13eqtri 2643 . . . . . . . . . 10 1𝑜 = ∅
1514fveq2i 6151 . . . . . . . . 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 4893 . . . . . . . . . 10 ⟨1𝑜, 𝑦⟩ ∈ V
1716rdg0 7462 . . . . . . . . 9 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘∅) = ⟨1𝑜, 𝑦
1815, 17eqtri 2643 . . . . . . . 8 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜) = ⟨1𝑜, 𝑦
1918fveq2i 6151 . . . . . . 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 2643 . . . . . 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 2673 . . . . 5 (𝑦𝑈 → ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
22 df-finxp 32850 . . . . . 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 2732 . . . . 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 697 . . . 4 (𝑦𝑈𝑦 ∈ (𝑈↑↑1𝑜))
251, 23mpbiran 952 . . . . 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 3189 . . . . . . 7 𝑦 ∈ V
2720eqcomi 2630 . . . . . . . . . 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 32856 . . . . . . . . . . . 12 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
2928neqned 2797 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
3029necomd 2845 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩) ≠ ∅)
3127, 30syl5eqner 2865 . . . . . . . . 9 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) ≠ ∅)
3231necomd 2845 . . . . . . . 8 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3332neneqd 2795 . . . . . . 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 705 . . . . . 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 2618 . 2 𝑈 = (𝑈↑↑1𝑜)
3938eqcomi 2630 1 (𝑈↑↑1𝑜) = 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  c0 3891  ifcif 4058  cop 4154   cuni 4402   × cxp 5072  Oncon0 5682  Lim wlim 5683  suc csuc 5684  cfv 5847  cmpt2 6606  ωcom 7012  1st c1st 7111  reccrdg 7450  1𝑜c1o 7498  ↑↑cfinxp 32849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-finxp 32850
This theorem is referenced by:  finxp2o  32865  finxp00  32868
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