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Theorem finxpsuclem 33364
Description: Lemma for finxpsuc 33365. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpsuclem.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpsuclem ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))
Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpsuclem
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano2 7128 . . . . . . . . . 10 (𝑁 ∈ ω → suc 𝑁 ∈ ω)
21adantr 480 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → suc 𝑁 ∈ ω)
3 1on 7612 . . . . . . . . . . . . 13 1𝑜 ∈ On
43onordi 5870 . . . . . . . . . . . 12 Ord 1𝑜
5 nnord 7115 . . . . . . . . . . . 12 (𝑁 ∈ ω → Ord 𝑁)
6 ordsseleq 5790 . . . . . . . . . . . 12 ((Ord 1𝑜 ∧ Ord 𝑁) → (1𝑜𝑁 ↔ (1𝑜𝑁 ∨ 1𝑜 = 𝑁)))
74, 5, 6sylancr 696 . . . . . . . . . . 11 (𝑁 ∈ ω → (1𝑜𝑁 ↔ (1𝑜𝑁 ∨ 1𝑜 = 𝑁)))
87biimpa 500 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (1𝑜𝑁 ∨ 1𝑜 = 𝑁))
9 elelsuc 5835 . . . . . . . . . . . . 13 (1𝑜𝑁 → 1𝑜 ∈ suc 𝑁)
109a1i 11 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1𝑜𝑁 → 1𝑜 ∈ suc 𝑁))
11 sucidg 5841 . . . . . . . . . . . . 13 (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12 eleq1 2718 . . . . . . . . . . . . 13 (1𝑜 = 𝑁 → (1𝑜 ∈ suc 𝑁𝑁 ∈ suc 𝑁))
1311, 12syl5ibrcom 237 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1𝑜 = 𝑁 → 1𝑜 ∈ suc 𝑁))
1410, 13jaod 394 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1𝑜𝑁 ∨ 1𝑜 = 𝑁) → 1𝑜 ∈ suc 𝑁))
1514adantr 480 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → ((1𝑜𝑁 ∨ 1𝑜 = 𝑁) → 1𝑜 ∈ suc 𝑁))
168, 15mpd 15 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → 1𝑜 ∈ suc 𝑁)
17 finxpsuclem.1 . . . . . . . . . 10 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1817finxpreclem6 33363 . . . . . . . . 9 ((suc 𝑁 ∈ ω ∧ 1𝑜 ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
192, 16, 18syl2anc 694 . . . . . . . 8 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
2019sselda 3636 . . . . . . 7 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
211ad2antrr 762 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22 df-2o 7606 . . . . . . . . . . . . . . 15 2𝑜 = suc 1𝑜
23 ordsucsssuc 7065 . . . . . . . . . . . . . . . . 17 ((Ord 1𝑜 ∧ Ord 𝑁) → (1𝑜𝑁 ↔ suc 1𝑜 ⊆ suc 𝑁))
244, 5, 23sylancr 696 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ω → (1𝑜𝑁 ↔ suc 1𝑜 ⊆ suc 𝑁))
2524biimpa 500 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → suc 1𝑜 ⊆ suc 𝑁)
2622, 25syl5eqss 3682 . . . . . . . . . . . . . 14 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → 2𝑜 ⊆ suc 𝑁)
2726adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2𝑜 ⊆ suc 𝑁)
28 simpr 476 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
2917finxpreclem4 33361 . . . . . . . . . . . . 13 (((suc 𝑁 ∈ ω ∧ 2𝑜 ⊆ suc 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁))
3021, 27, 28, 29syl21anc 1365 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁))
31 ordunisuc 7074 . . . . . . . . . . . . . . . 16 (Ord 𝑁 suc 𝑁 = 𝑁)
325, 31syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → suc 𝑁 = 𝑁)
33 opeq1 4433 . . . . . . . . . . . . . . . 16 ( suc 𝑁 = 𝑁 → ⟨ suc 𝑁, (1st𝑦)⟩ = ⟨𝑁, (1st𝑦)⟩)
34 rdgeq2 7553 . . . . . . . . . . . . . . . 16 (⟨ suc 𝑁, (1st𝑦)⟩ = ⟨𝑁, (1st𝑦)⟩ → rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st𝑦)⟩))
3533, 34syl 17 . . . . . . . . . . . . . . 15 ( suc 𝑁 = 𝑁 → rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st𝑦)⟩))
3632, 35syl 17 . . . . . . . . . . . . . 14 (𝑁 ∈ ω → rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st𝑦)⟩))
3736, 32fveq12d 6235 . . . . . . . . . . . . 13 (𝑁 ∈ ω → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3837ad2antrr 762 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3930, 38eqtrd 2685 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
4039eqeq2d 2661 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
411biantrurd 528 . . . . . . . . . . . 12 (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
4217dffinxpf 33352 . . . . . . . . . . . . 13 (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
4342abeq2i 2764 . . . . . . . . . . . 12 (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
4441, 43syl6rbbr 279 . . . . . . . . . . 11 (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
4544ad2antrr 762 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46 fvex 6239 . . . . . . . . . . . . 13 (1st𝑦) ∈ V
47 opeq2 4434 . . . . . . . . . . . . . . . . 17 (𝑧 = (1st𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st𝑦)⟩)
48 rdgeq2 7553 . . . . . . . . . . . . . . . . 17 (⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st𝑦)⟩ → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st𝑦)⟩))
4947, 48syl 17 . . . . . . . . . . . . . . . 16 (𝑧 = (1st𝑦) → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st𝑦)⟩))
5049fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑧 = (1st𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
5150eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5251anbi2d 740 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))))
5317dffinxpf 33352 . . . . . . . . . . . . 13 (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
5446, 52, 53elab2 3386 . . . . . . . . . . . 12 ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5554baib 964 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5655ad2antrr 762 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5740, 45, 563bitr4d 300 . . . . . . . . 9 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st𝑦) ∈ (𝑈↑↑𝑁)))
5857biimpd 219 . . . . . . . 8 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
5958impancom 455 . . . . . . 7 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (𝑦 ∈ (V × 𝑈) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
6020, 59mpd 15 . . . . . 6 (((𝑁 ∈ ω ∧ 1𝑜𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (1st𝑦) ∈ (𝑈↑↑𝑁))
6160ex 449 . . . . 5 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
6220ex 449 . . . . 5 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
6361, 62jcad 554 . . . 4 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
6457exbiri 651 . . . . . 6 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
6564impd 446 . . . . 5 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
6665ancomsd 469 . . . 4 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
6763, 66impbid 202 . . 3 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68 elxp8 33349 . . 3 (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
6967, 68syl6bbr 278 . 2 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
7069eqrdv 2649 1 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  cop 4216   cuni 4468   × cxp 5141  Ord word 5760  suc csuc 5763  cfv 5926  cmpt2 6692  ωcom 7107  1st c1st 7208  reccrdg 7550  1𝑜c1o 7598  2𝑜c2o 7599  ↑↑cfinxp 33350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-finxp 33351
This theorem is referenced by:  finxpsuc  33365
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