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Theorem finxpreclem6 32865
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem5.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem6 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
Distinct variable groups:   𝑥,𝑛,𝑁   𝑈,𝑛,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem6
Dummy variables 𝑚 𝑜 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2686 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2 eleq2 2687 . . . . 5 (𝑛 = 𝑁 → (1𝑜𝑛 ↔ 1𝑜𝑁))
31, 2anbi12d 746 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1𝑜𝑛) ↔ (𝑁 ∈ ω ∧ 1𝑜𝑁)))
4 anass 680 . . . . . . . . 9 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
5 nfv 1840 . . . . . . . . . . . . . . 15 𝑥(𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))
6 finxpreclem5.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7 nfmpt22 6676 . . . . . . . . . . . . . . . . . . . 20 𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
86, 7nfcxfr 2759 . . . . . . . . . . . . . . . . . . 19 𝑥𝐹
9 nfcv 2761 . . . . . . . . . . . . . . . . . . 19 𝑥𝑛, 𝑦
108, 9nfrdg 7455 . . . . . . . . . . . . . . . . . 18 𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11 nfcv 2761 . . . . . . . . . . . . . . . . . 18 𝑥𝑛
1210, 11nffv 6155 . . . . . . . . . . . . . . . . 17 𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
1312nfeq2 2776 . . . . . . . . . . . . . . . 16 𝑥∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
1413nfn 1781 . . . . . . . . . . . . . . 15 𝑥 ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
155, 14nfim 1822 . . . . . . . . . . . . . 14 𝑥((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
16 eleq1 2686 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈)))
1716notbid 308 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (V × 𝑈) ↔ ¬ 𝑦 ∈ (V × 𝑈)))
1817anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
1918anbi2d 739 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) ↔ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))))
20 opeq2 4371 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21 rdgeq2 7453 . . . . . . . . . . . . . . . . . . 19 (⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
2220, 21syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
2322fveq1d 6150 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
2423eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2524notbid 308 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
2619, 25imbi12d 334 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛)) ↔ ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
27 anass 680 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28 vex 3189 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
29 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = ∅ → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅))
3029eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
31 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑜 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜))
3231eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
33 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = suc 𝑜 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜))
3433eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
35 opex 4893 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛, 𝑥⟩ ∈ V
3635rdg0 7462 . . . . . . . . . . . . . . . . . . . . . . . 24 (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥
3736a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩)
38 nnon 7018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑜 ∈ ω → 𝑜 ∈ On)
39 rdgsuc 7465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑜 ∈ On → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
4038, 39syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑜 ∈ ω → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
41 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
4240, 41sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
436finxpreclem5 32864 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ 1𝑜𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
4443imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
4542, 44sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) ∧ ((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)
4645expl 647 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑜 ∈ ω → (((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ ∧ ((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
4746expcomd 454 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑜 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)))
4830, 32, 34, 37, 47finds2 7041 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
49 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
50 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚))
5150eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
5251imbi2d 330 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩) ↔ (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)))
5349, 52imbi12d 334 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → ((𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) ↔ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))))
5448, 53mpbiri 248 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5554equcoms 1944 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
5628, 55vtocle 3268 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5727, 56syl5bir 233 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
5857anabsi5 857 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
59 vex 3189 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
6028, 59opnzi 4903 . . . . . . . . . . . . . . . . . 18 𝑛, 𝑥⟩ ≠ ∅
6160a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
6258, 61eqnetrd 2857 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
6362necomd 2845 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6463neneqd 2795 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
6515, 26, 64chvar 2261 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
6665intnand 961 . . . . . . . . . . . 12 ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
6766adantl 482 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
68 abid 2609 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
69 opeq1 4370 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
70 rdgeq2 7453 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
7169, 70syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
72 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁𝑛 = 𝑁)
7371, 72fveq12d 6154 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
7473eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))
751, 74anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))))
7675abbidv 2738 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))})
776dffinxpf 32854 . . . . . . . . . . . . . . 15 (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
7876, 77syl6eqr 2673 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
7978eleq2d 2684 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
8068, 79syl5rbbr 275 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
8180adantr 481 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
8267, 81mtbird 315 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
8382ex 450 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1𝑜𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
844, 83syl5bi 232 . . . . . . . 8 (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
8584expdimp 453 . . . . . . 7 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1𝑜𝑛)) → (¬ 𝑦 ∈ (V × 𝑈) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
8685con4d 114 . . . . . 6 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1𝑜𝑛)) → (𝑦 ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (V × 𝑈)))
8786ssrdv 3589 . . . . 5 ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1𝑜𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
8887ex 450 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1𝑜𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
893, 88sylbird 250 . . 3 (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
9089vtocleg 3265 . 2 (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
9190anabsi5 857 1 ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  Vcvv 3186  wss 3555  c0 3891  ifcif 4058  cop 4154   cuni 4402   × cxp 5072  Oncon0 5682  suc csuc 5684  cfv 5847  cmpt2 6606  ωcom 7012  1st c1st 7111  reccrdg 7450  1𝑜c1o 7498  ↑↑cfinxp 32852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-finxp 32853
This theorem is referenced by:  finxpsuclem  32866
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