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Theorem finxpreclem4 34206
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 8116 . . . . . . . 8 2o ∈ ω
2 nnon 7442 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 7962 . . . . . . . . . . . . . 14 2o ∈ On
4 oawordeu 8031 . . . . . . . . . . . . . 14 (((2o ∈ On ∧ 𝑁 ∈ On) ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
53, 4mpanl1 696 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
6 riotasbc 6992 . . . . . . . . . . . . 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 6981 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V
9 sbceq1g 4286 . . . . . . . . . . . . . 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 7061 . . . . . . . . . . . . . . . 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 4299 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2o +o 𝑜) = 𝑁)
1413oveq2i 7027 . . . . . . . . . . . . . . 15 (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1512, 14eqtri 2819 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1615eqeq1i 2800 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
1710, 16bitri 276 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
187, 17sylib 219 . . . . . . . . . . 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 2883 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
22 riotacl 6991 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
23 riotaund 7013 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) = ∅)
24 0elon 6119 . . . . . . . . . . . 12 ∅ ∈ On
2523, 24syl6eqel 2891 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
2622, 25pm2.61i 183 . . . . . . . . . 10 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On
27 nnarcl 8092 . . . . . . . . . . . 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 686 . . . . . . . . . . 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, 29syl6bbr 290 . . . . . . . . . 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 219 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2o𝑁) → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
33 nnacom 8093 . . . . . . . 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 7954 . . . . . . . . 9 2o = suc 1o
3635oveq2i 7027 . . . . . . . 8 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o)
37 1onn 8115 . . . . . . . . 9 1o ∈ ω
38 nnasuc 8082 . . . . . . . . 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, 39syl5eq 2843 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
4134, 19, 403eqtr3d 2839 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
422adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ On)
43 sucidg 6144 . . . . . . . . . . . 12 (1o ∈ ω → 1o ∈ suc 1o)
4437, 43ax-mp 5 . . . . . . . . . . 11 1o ∈ suc 1o
4544, 35eleqtrri 2882 . . . . . . . . . 10 1o ∈ 2o
46 ssel 3883 . . . . . . . . . 10 (2o𝑁 → (1o ∈ 2o → 1o𝑁))
4745, 46mpi 20 . . . . . . . . 9 (2o𝑁 → 1o𝑁)
4847ne0d 4221 . . . . . . . 8 (2o𝑁𝑁 ≠ ∅)
4948adantl 482 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ≠ ∅)
50 nnlim 7449 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5150adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ¬ Lim 𝑁)
52 onsucuni3 34179 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5342, 49, 51, 52syl3anc 1364 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc 𝑁)
54 nnacom 8093 . . . . . . . 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 6131 . . . . . . 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 2839 . . . . 5 ((𝑁 ∈ ω ∧ 2o𝑁) → suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
59 ordom 7445 . . . . . . . . 9 Ord ω
60 ordelss 6082 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6159, 60mpan 686 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
62 nnfi 8557 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
63 nnunifi 8615 . . . . . . . 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 8087 . . . . . . 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 7460 . . . . . 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 233 . . . 4 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
7170fveq2d 6542 . . 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 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7574finxpreclem3 34205 . . . . . 6 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
76 df-1o 7953 . . . . . . . 8 1o = suc ∅
7776fveq2i 6541 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
78 rdgsuc 7912 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
7924, 78ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
80 opex 5248 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
8180rdg0 7909 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8281fveq2i 6541 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8377, 79, 823eqtri 2823 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (𝐹‘⟨𝑁, 𝑦⟩)
8475, 83syl6reqr 2850 . . . . 5 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = ⟨ 𝑁, (1st𝑦)⟩)
8584fveq2d 6542 . . . 4 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
86 2on0 7964 . . . . . 6 2o ≠ ∅
87 nnlim 7449 . . . . . . 7 (2o ∈ ω → ¬ Lim 2o)
881, 87ax-mp 5 . . . . . 6 ¬ Lim 2o
89 rdgsucuni 34181 . . . . . 6 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)))
903, 86, 88, 89mp3an 1453 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
91 1oequni2o 34180 . . . . . . 7 1o = 2o
9291fveq2i 6541 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)
9392fveq2i 6541 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
9490, 93eqtr4i 2822 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o))
9576fveq2i 6541 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
96 rdgsuc 7912 . . . . . 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 5248 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
9998rdg0 7909 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
10099fveq2i 6541 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10195, 97, 1003eqtri 2823 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10285, 94, 1013eqtr4g 2856 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o))
103 1on 7960 . . . 4 1o ∈ On
104 rdgeqoa 34182 . . . 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 1443 . . 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 6542 . . 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 2838 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 207  wa 396   = wceq 1522  wcel 2081  wne 2984  ∃!wreu 3107  Vcvv 3437  [wsbc 3706  csb 3811  wss 3859  c0 4211  ifcif 4381  cop 4478   cuni 4745   × cxp 5441  Ord word 6065  Oncon0 6066  Lim wlim 6067  suc csuc 6068  cfv 6225  crio 6976  (class class class)co 7016  cmpo 7018  ωcom 7436  1st c1st 7543  reccrdg 7897  1oc1o 7946  2oc2o 7947   +o coa 7950  Fincfn 8357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361
This theorem is referenced by:  finxpsuclem  34209
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