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Theorem finxp1o 35863
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 8586 . . . . . 6 1o ∈ ω
21a1i 11 . . . . 5 (𝑦𝑈 → 1o ∈ ω)
3 finxpreclem1 35860 . . . . . 6 (𝑦𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
4 1on 8424 . . . . . . . 8 1o ∈ On
5 1n0 8434 . . . . . . . 8 1o ≠ ∅
6 nnlim 7816 . . . . . . . . 9 (1o ∈ ω → ¬ Lim 1o)
71, 6ax-mp 5 . . . . . . . 8 ¬ Lim 1o
8 rdgsucuni 35840 . . . . . . . 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 1461 . . . . . . 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 8412 . . . . . . . . . . . 12 1o = suc ∅
1110unieqi 4878 . . . . . . . . . . 11 1o = suc ∅
12 0elon 6371 . . . . . . . . . . . 12 ∅ ∈ On
1312onunisuci 6437 . . . . . . . . . . 11 suc ∅ = ∅
1411, 13eqtri 2764 . . . . . . . . . 10 1o = ∅
1514fveq2i 6845 . . . . . . . . 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 5421 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
1716rdg0 8367 . . . . . . . . 9 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘∅) = ⟨1o, 𝑦
1815, 17eqtri 2764 . . . . . . . 8 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘ 1o) = ⟨1o, 𝑦
1918fveq2i 6845 . . . . . . 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 2764 . . . . . 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, 20eqtr4di 2794 . . . . 5 (𝑦𝑈 → ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
22 df-finxp 35855 . . . . . 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))}
2322eqabi 2881 . . . . 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 583 . . . 4 (𝑦𝑈𝑦 ∈ (𝑈↑↑1o))
251, 23mpbiran 707 . . . . 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 3449 . . . . . . 7 𝑦 ∈ V
2720eqcomi 2745 . . . . . . . . . 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 35861 . . . . . . . . . . . 12 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
2928neqned 2950 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩))
3029necomd 2999 . . . . . . . . . 10 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑦⟩) ≠ ∅)
3127, 30eqnetrrid 3019 . . . . . . . . 9 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) ≠ ∅)
3231necomd 2999 . . . . . . . 8 ((𝑦 ∈ V ∧ ¬ 𝑦𝑈) → ∅ ≠ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
3332neneqd 2948 . . . . . . 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 688 . . . . . 6 𝑦𝑈 → ¬ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o))
3534con4i 114 . . . . 5 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨1o, 𝑦⟩)‘1o) → 𝑦𝑈)
3625, 35sylbi 216 . . . 4 (𝑦 ∈ (𝑈↑↑1o) → 𝑦𝑈)
3724, 36impbii 208 . . 3 (𝑦𝑈𝑦 ∈ (𝑈↑↑1o))
3837eqriv 2733 . 2 𝑈 = (𝑈↑↑1o)
3938eqcomi 2745 1 (𝑈↑↑1o) = 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  c0 4282  ifcif 4486  cop 4592   cuni 4865   × cxp 5631  Oncon0 6317  Lim wlim 6318  suc csuc 6319  cfv 6496  cmpo 7359  ωcom 7802  1st c1st 7919  reccrdg 8355  1oc1o 8405  ↑↑cfinxp 35854
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-finxp 35855
This theorem is referenced by:  finxp2o  35870  finxp00  35873
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