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Theorem finxpsuclem 34830
 Description: Lemma for finxpsuc 34831. (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 7584 . . . . . . . . . 10 (𝑁 ∈ ω → suc 𝑁 ∈ ω)
21adantr 484 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 1o𝑁) → suc 𝑁 ∈ ω)
3 1on 8094 . . . . . . . . . . . . 13 1o ∈ On
43onordi 6263 . . . . . . . . . . . 12 Ord 1o
5 nnord 7570 . . . . . . . . . . . 12 (𝑁 ∈ ω → Ord 𝑁)
6 ordsseleq 6188 . . . . . . . . . . . 12 ((Ord 1o ∧ Ord 𝑁) → (1o𝑁 ↔ (1o𝑁 ∨ 1o = 𝑁)))
74, 5, 6sylancr 590 . . . . . . . . . . 11 (𝑁 ∈ ω → (1o𝑁 ↔ (1o𝑁 ∨ 1o = 𝑁)))
87biimpa 480 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 1o𝑁) → (1o𝑁 ∨ 1o = 𝑁))
9 elelsuc 6231 . . . . . . . . . . . . 13 (1o𝑁 → 1o ∈ suc 𝑁)
109a1i 11 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1o𝑁 → 1o ∈ suc 𝑁))
11 sucidg 6237 . . . . . . . . . . . . 13 (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12 eleq1 2877 . . . . . . . . . . . . 13 (1o = 𝑁 → (1o ∈ suc 𝑁𝑁 ∈ suc 𝑁))
1311, 12syl5ibrcom 250 . . . . . . . . . . . 12 (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
1410, 13jaod 856 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1o𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
1514adantr 484 . . . . . . . . . 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 34829 . . . . . . . . 9 ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
192, 16, 18syl2anc 587 . . . . . . . 8 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
2019sselda 3915 . . . . . . 7 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
211ad2antrr 725 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22 df-2o 8088 . . . . . . . . . . . . . . 15 2o = suc 1o
23 ordsucsssuc 7520 . . . . . . . . . . . . . . . . 17 ((Ord 1o ∧ Ord 𝑁) → (1o𝑁 ↔ suc 1o ⊆ suc 𝑁))
244, 5, 23sylancr 590 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ω → (1o𝑁 ↔ suc 1o ⊆ suc 𝑁))
2524biimpa 480 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ω ∧ 1o𝑁) → suc 1o ⊆ suc 𝑁)
2622, 25eqsstrid 3963 . . . . . . . . . . . . . 14 ((𝑁 ∈ ω ∧ 1o𝑁) → 2o ⊆ suc 𝑁)
2726adantr 484 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28 simpr 488 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
2917finxpreclem4 34827 . . . . . . . . . . . . 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 7529 . . . . . . . . . . . . . . . 16 (Ord 𝑁 suc 𝑁 = 𝑁)
325, 31syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ ω → suc 𝑁 = 𝑁)
33 opeq1 4763 . . . . . . . . . . . . . . . 16 ( suc 𝑁 = 𝑁 → ⟨ suc 𝑁, (1st𝑦)⟩ = ⟨𝑁, (1st𝑦)⟩)
34 rdgeq2 8033 . . . . . . . . . . . . . . . 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 6652 . . . . . . . . . . . . 13 (𝑁 ∈ ω → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3837ad2antrr 725 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ suc 𝑁, (1st𝑦)⟩)‘ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
3930, 38eqtrd 2833 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
4039eqeq2d 2809 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
4117dffinxpf 34818 . . . . . . . . . . . . 13 (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
4241abeq2i 2925 . . . . . . . . . . . 12 (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
431biantrurd 536 . . . . . . . . . . . 12 (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
4442, 43bitr4id 293 . . . . . . . . . . 11 (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
4544ad2antrr 725 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46 fvex 6658 . . . . . . . . . . . . 13 (1st𝑦) ∈ V
47 opeq2 4765 . . . . . . . . . . . . . . . . 17 (𝑧 = (1st𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st𝑦)⟩)
48 rdgeq2 8033 . . . . . . . . . . . . . . . . 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 6647 . . . . . . . . . . . . . . 15 (𝑧 = (1st𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))
5150eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5251anbi2d 631 . . . . . . . . . . . . 13 (𝑧 = (1st𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁))))
5317dffinxpf 34818 . . . . . . . . . . . . 13 (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
5446, 52, 53elab2 3618 . . . . . . . . . . . 12 ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5554baib 539 . . . . . . . . . . 11 (𝑁 ∈ ω → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5655ad2antrr 725 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st𝑦)⟩)‘𝑁)))
5740, 45, 563bitr4d 314 . . . . . . . . 9 (((𝑁 ∈ ω ∧ 1o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st𝑦) ∈ (𝑈↑↑𝑁)))
5857biimpd 232 . . . . . . . 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 416 . . . . 5 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st𝑦) ∈ (𝑈↑↑𝑁)))
6220ex 416 . . . . 5 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
6361, 62jcad 516 . . . 4 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
6457exbiri 810 . . . . . 6 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
6564impd 414 . . . . 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 215 . . 3 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68 elxp8 34804 . . 3 (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
6967, 68bitr4di 292 . 2 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
7069eqrdv 2796 1 ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ifcif 4425  ⟨cop 4531  ∪ cuni 4800   × cxp 5517  Ord word 6158  suc csuc 6161  ‘cfv 6324   ∈ cmpo 7137  ωcom 7562  1st c1st 7671  reccrdg 8030  1oc1o 8080  2oc2o 8081  ↑↑cfinxp 34816 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-finxp 34817 This theorem is referenced by:  finxpsuc  34831
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