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Theorem finxpreclem4 36367
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem4.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem4 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑦,𝑛,𝑥
Allowed substitution hints:   𝑈(𝑦)   𝐹(𝑥,𝑦,𝑛)   𝑁(𝑦)

Proof of Theorem finxpreclem4
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 2onn 8643 . . . . . . . 8 2o ∈ ω
2 nnon 7863 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 8482 . . . . . . . . . . . . . 14 2o ∈ On
4 oawordeu 8557 . . . . . . . . . . . . . 14 (((2o ∈ On ∧ 𝑁 ∈ On) ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
53, 4mpanl1 698 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
6 riotasbc 7386 . . . . . . . . . . . . 13 (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁[(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
75, 6syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ On ∧ 2o𝑁) → [(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
8 riotaex 7371 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V
9 sbceq1g 4414 . . . . . . . . . . . . . 14 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V → ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁))
108, 9ax-mp 5 . . . . . . . . . . . . 13 ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁)
11 csbov2g 7457 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜))
128, 11ax-mp 5 . . . . . . . . . . . . . . 15 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜)
138csbvargi 4432 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2o +o 𝑜) = 𝑁)
1413oveq2i 7422 . . . . . . . . . . . . . . 15 (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1512, 14eqtri 2760 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1615eqeq1i 2737 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
1710, 16bitri 274 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
187, 17sylib 217 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
192, 18sylan 580 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
20 simpl 483 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ ω)
2119, 20eqeltrd 2833 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
22 riotacl 7385 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
23 riotaund 7407 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) = ∅)
24 0elon 6418 . . . . . . . . . . . 12 ∅ ∈ On
2523, 24eqeltrdi 2841 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
2622, 25pm2.61i 182 . . . . . . . . . 10 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On
27 nnarcl 8618 . . . . . . . . . . . 12 ((2o ∈ On ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On) → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)))
283, 27mpan 688 . . . . . . . . . . 11 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)))
291biantrur 531 . . . . . . . . . . 11 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω))
3028, 29bitr4di 288 . . . . . . . . . 10 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω))
3126, 30ax-mp 5 . . . . . . . . 9 ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
3221, 31sylib 217 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2o𝑁) → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
33 nnacom 8619 . . . . . . . 8 ((2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o))
341, 32, 33sylancr 587 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o))
35 df-2o 8469 . . . . . . . . 9 2o = suc 1o
3635oveq2i 7422 . . . . . . . 8 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o)
37 1onn 8641 . . . . . . . . 9 1o ∈ ω
38 nnasuc 8608 . . . . . . . . 9 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ∧ 1o ∈ ω) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
3932, 37, 38sylancl 586 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
4036, 39eqtrid 2784 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
4134, 19, 403eqtr3d 2780 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
422adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ On)
43 sucidg 6445 . . . . . . . . . . . 12 (1o ∈ ω → 1o ∈ suc 1o)
4437, 43ax-mp 5 . . . . . . . . . . 11 1o ∈ suc 1o
4544, 35eleqtrri 2832 . . . . . . . . . 10 1o ∈ 2o
46 ssel 3975 . . . . . . . . . 10 (2o𝑁 → (1o ∈ 2o → 1o𝑁))
4745, 46mpi 20 . . . . . . . . 9 (2o𝑁 → 1o𝑁)
4847ne0d 4335 . . . . . . . 8 (2o𝑁𝑁 ≠ ∅)
4948adantl 482 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ≠ ∅)
50 nnlim 7871 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5150adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ¬ Lim 𝑁)
52 onsucuni3 36340 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5342, 49, 51, 52syl3anc 1371 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc 𝑁)
54 nnacom 8619 . . . . . . . 8 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ∧ 1o ∈ ω) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5532, 37, 54sylancl 586 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
56 suceq 6430 . . . . . . 7 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) → suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5755, 56syl 17 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5841, 53, 573eqtr3d 2780 . . . . 5 ((𝑁 ∈ ω ∧ 2o𝑁) → suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
59 ordom 7867 . . . . . . . . 9 Ord ω
60 ordelss 6380 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6159, 60mpan 688 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
62 nnfi 9169 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
63 nnunifi 9296 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6461, 62, 63syl2anc 584 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6564adantr 481 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ ω)
66 nnacl 8613 . . . . . . 7 ((1o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
6737, 32, 66sylancr 587 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
68 peano4 7885 . . . . . 6 (( 𝑁 ∈ ω ∧ (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω) → (suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ↔ 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
6965, 67, 68syl2anc 584 . . . . 5 ((𝑁 ∈ ω ∧ 2o𝑁) → (suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ↔ 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7058, 69mpbid 231 . . . 4 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
7170fveq2d 6895 . . 3 ((𝑁 ∈ ω ∧ 2o𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7271adantr 481 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7332adantr 481 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
74 df-1o 8468 . . . . . . . 8 1o = suc ∅
7574fveq2i 6894 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
76 rdgsuc 8426 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
7724, 76ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
78 opex 5464 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
7978rdg0 8423 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8079fveq2i 6894 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8175, 77, 803eqtri 2764 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (𝐹‘⟨𝑁, 𝑦⟩)
82 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
8382finxpreclem3 36366 . . . . . 6 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
8481, 83eqtr4id 2791 . . . . 5 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = ⟨ 𝑁, (1st𝑦)⟩)
8584fveq2d 6895 . . . 4 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
86 2on0 8484 . . . . . 6 2o ≠ ∅
87 nnlim 7871 . . . . . . 7 (2o ∈ ω → ¬ Lim 2o)
881, 87ax-mp 5 . . . . . 6 ¬ Lim 2o
89 rdgsucuni 36342 . . . . . 6 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)))
903, 86, 88, 89mp3an 1461 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
91 1oequni2o 36341 . . . . . . 7 1o = 2o
9291fveq2i 6894 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)
9392fveq2i 6894 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
9490, 93eqtr4i 2763 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o))
9574fveq2i 6894 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
96 rdgsuc 8426 . . . . . 6 (∅ ∈ On → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)))
9724, 96ax-mp 5 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅))
98 opex 5464 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
9998rdg0 8423 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
10099fveq2i 6894 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10195, 97, 1003eqtri 2764 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10285, 94, 1013eqtr4g 2797 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o))
103 1on 8480 . . . 4 1o ∈ On
104 rdgeqoa 36343 . . . 4 ((2o ∈ On ∧ 1o ∈ On ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))))
1053, 103, 104mp3an12 1451 . . 3 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))))
10673, 102, 105sylc 65 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
10719fveq2d 6895 . . 3 ((𝑁 ∈ ω ∧ 2o𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
108107adantr 481 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
10972, 106, 1083eqtr2rd 2779 1 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  ∃!wreu 3374  Vcvv 3474  [wsbc 3777  csb 3893  wss 3948  c0 4322  ifcif 4528  cop 4634   cuni 4908   × cxp 5674  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  cfv 6543  crio 7366  (class class class)co 7411  cmpo 7413  ωcom 7857  1st c1st 7975  reccrdg 8411  1oc1o 8461  2oc2o 8462   +o coa 8465  Fincfn 8941
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-en 8942  df-fin 8945
This theorem is referenced by:  finxpsuclem  36370
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