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Theorem finxp1o 32405
 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 7606 . . . . . 6 1𝑜 ∈ ω
21a1i 11 . . . . 5 (𝑦𝑈 → 1𝑜 ∈ ω)
3 finxpreclem1 32402 . . . . . 6 (𝑦𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
4 1on 7454 . . . . . . . 8 1𝑜 ∈ On
5 1n0 7462 . . . . . . . 8 1𝑜 ≠ ∅
6 nnlim 6970 . . . . . . . . 9 (1𝑜 ∈ ω → ¬ Lim 1𝑜)
71, 6ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
8 rdgsucuni 32393 . . . . . . . 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 1416 . . . . . . 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 7447 . . . . . . . . . . . 12 1𝑜 = suc ∅
1110unieqi 4381 . . . . . . . . . . 11 1𝑜 = suc ∅
12 0elon 5695 . . . . . . . . . . . 12 ∅ ∈ On
1312onunisuci 5758 . . . . . . . . . . 11 suc ∅ = ∅
1411, 13eqtri 2632 . . . . . . . . . 10 1𝑜 = ∅
1514fveq2i 6106 . . . . . . . . 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 4859 . . . . . . . . . 10 ⟨1𝑜, 𝑦⟩ ∈ V
1716rdg0 7404 . . . . . . . . 9 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘∅) = ⟨1𝑜, 𝑦
1815, 17eqtri 2632 . . . . . . . 8 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘ 1𝑜) = ⟨1𝑜, 𝑦
1918fveq2i 6106 . . . . . . 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 2632 . . . . . 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 2662 . . . . 5 (𝑦𝑈 → ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
22 df-finxp 32397 . . . . . 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 2722 . . . . 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 695 . . . 4 (𝑦𝑈𝑦 ∈ (𝑈↑↑1𝑜))
251, 23mpbiran 955 . . . . 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 3176 . . . . . . 7 𝑦 ∈ V
2720eqcomi 2619 . . . . . . . . . 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 32403 . . . . . . . . . . . 12 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
2928neqned 2789 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩))
3029necomd 2837 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑦⟩) ≠ ∅)
3127, 30syl5eqner 2857 . . . . . . . . 9 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) ≠ ∅)
3231necomd 2837 . . . . . . . 8 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3332neneqd 2787 . . . . . . 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 702 . . . . . 6 𝑦𝑈 → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜))
3534con4i 112 . . . . 5 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1𝑜, 𝑦⟩)‘1𝑜) → 𝑦𝑈)
3625, 35sylbi 206 . . . 4 (𝑦 ∈ (𝑈↑↑1𝑜) → 𝑦𝑈)
3724, 36impbii 198 . . 3 (𝑦𝑈𝑦 ∈ (𝑈↑↑1𝑜))
3837eqriv 2607 . 2 𝑈 = (𝑈↑↑1𝑜)
3938eqcomi 2619 1 (𝑈↑↑1𝑜) = 𝑈
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ifcif 4036  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  Oncon0 5640  Lim wlim 5641  suc csuc 5642  ‘cfv 5804   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  reccrdg 7392  1𝑜c1o 7440  ↑↑cfinxp 32396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-finxp 32397 This theorem is referenced by:  finxp2o  32412  finxp00  32415
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