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Theorem finxpsuclem 35868
Description: Lemma for finxpsuc 35869. (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 7827 . . . . . . . . . 10 (𝑁 ∈ ω → suc 𝑁 ∈ ω)
21adantr 481 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 1o𝑁) → suc 𝑁 ∈ ω)
3 1on 8424 . . . . . . . . . . . . 13 1o ∈ On
43onordi 6428 . . . . . . . . . . . 12 Ord 1o
5 nnord 7810 . . . . . . . . . . . 12 (𝑁 ∈ ω → Ord 𝑁)
6 ordsseleq 6346 . . . . . . . . . . . 12 ((Ord 1o ∧ Ord 𝑁) → (1o𝑁 ↔ (1o𝑁 ∨ 1o = 𝑁)))
74, 5, 6sylancr 587 . . . . . . . . . . 11 (𝑁 ∈ ω → (1o𝑁 ↔ (1o𝑁 ∨ 1o = 𝑁)))
87biimpa 477 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 1o𝑁) → (1o𝑁 ∨ 1o = 𝑁))
9 elelsuc 6390 . . . . . . . . . . . . 13 (1o𝑁 → 1o ∈ suc 𝑁)
109a1i 11 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1o𝑁 → 1o ∈ suc 𝑁))
11 sucidg 6398 . . . . . . . . . . . . 13 (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12 eleq1 2825 . . . . . . . . . . . . 13 (1o = 𝑁 → (1o ∈ suc 𝑁𝑁 ∈ suc 𝑁))
1311, 12syl5ibrcom 246 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
1410, 13jaod 857 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1o𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
1514adantr 481 . . . . . . . . . 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 35867 . . . . . . . . 9 ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
192, 16, 18syl2anc 584 . . . . . . . 8 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
2019sselda 3944 . . . . . . 7 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
211ad2antrr 724 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22 df-2o 8413 . . . . . . . . . . . . . . 15 2o = suc 1o
23 ordsucsssuc 7758 . . . . . . . . . . . . . . . . 17 ((Ord 1o ∧ Ord 𝑁) → (1o𝑁 ↔ suc 1o ⊆ suc 𝑁))
244, 5, 23sylancr 587 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ω → (1o𝑁 ↔ suc 1o ⊆ suc 𝑁))
2524biimpa 477 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ω ∧ 1o𝑁) → suc 1o ⊆ suc 𝑁)
2622, 25eqsstrid 3992 . . . . . . . . . . . . . 14 ((𝑁 ∈ ω ∧ 1o𝑁) → 2o ⊆ suc 𝑁)
2726adantr 481 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28 simpr 485 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
2917finxpreclem4 35865 . . . . . . . . . . . . 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 836 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁))
31 ordunisuc 7767 . . . . . . . . . . . . . . . 16 (Ord 𝑁 suc 𝑁 = 𝑁)
325, 31syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → suc 𝑁 = 𝑁)
33 opeq1 4830 . . . . . . . . . . . . . . . 16 ( suc 𝑁 = 𝑁 → ⟨ suc 𝑁, (1st𝑦)⟩ = ⟨𝑁, (1st𝑦)⟩)
34 rdgeq2 8358 . . . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . 13 (𝑁 ∈ ω → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3837ad2antrr 724 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3930, 38eqtrd 2776 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
4039eqeq2d 2747 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
4117dffinxpf 35856 . . . . . . . . . . . . 13 (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
4241eqabi 2881 . . . . . . . . . . . 12 (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
431biantrurd 533 . . . . . . . . . . . 12 (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
4442, 43bitr4id 289 . . . . . . . . . . 11 (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
4544ad2antrr 724 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46 fvex 6855 . . . . . . . . . . . . 13 (1st𝑦) ∈ V
47 opeq2 4831 . . . . . . . . . . . . . . . . 17 (𝑧 = (1st𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st𝑦)⟩)
48 rdgeq2 8358 . . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . 15 (𝑧 = (1st𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
5150eqeq2d 2747 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5251anbi2d 629 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))))
5317dffinxpf 35856 . . . . . . . . . . . . 13 (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
5446, 52, 53elab2 3634 . . . . . . . . . . . 12 ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5554baib 536 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5655ad2antrr 724 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5740, 45, 563bitr4d 310 . . . . . . . . 9 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st𝑦) ∈ (𝑈↑↑𝑁)))
5857biimpd 228 . . . . . . . 8 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
5958impancom 452 . . . . . . 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 413 . . . . 5 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
6220ex 413 . . . . 5 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
6361, 62jcad 513 . . . 4 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
6457exbiri 809 . . . . . 6 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
6564impd 411 . . . . 5 ((𝑁 ∈ ω ∧ 1o𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
6665ancomsd 466 . . . 4 ((𝑁 ∈ ω ∧ 1o𝑁) → (((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
6763, 66impbid 211 . . 3 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68 elxp8 35842 . . 3 (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
6967, 68bitr4di 288 . 2 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
7069eqrdv 2734 1 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  c0 4282  ifcif 4486  cop 4592   cuni 4865   × cxp 5631  Ord word 6316  suc csuc 6319  cfv 6496  cmpo 7359  ωcom 7802  1st c1st 7919  reccrdg 8355  1oc1o 8405  2oc2o 8406  ↑↑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-rmo 3353  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-int 4908  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-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-en 8884  df-fin 8887  df-finxp 35855
This theorem is referenced by:  finxpsuc  35869
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