Theorem finxpreclem3 35564

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem3.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem3	⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem3

Step	Hyp	Ref	Expression
1		finxpreclem3.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
2	1	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
3		eqeq1 2742	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1o ↔ 𝑁 = 1o))
4		eleq1 2826	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
5	3, 4	bi2anan9 636	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈)))
6		eleq1 2826	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
7	6	adantl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8		unieq 4850	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
9	8	adantr 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
10		fveq2 6774	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1st ‘𝑥) = (1st ‘𝑋))
11	10	adantl 482	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1st ‘𝑥) = (1st ‘𝑋))
12	9, 11	opeq12d 4812	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
13		opeq12 4806	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
14	7, 12, 13	ifbieq12d 4487	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
15	5, 14	ifbieq2d 4485	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)))
16		sssucid 6343	. . . . . . . . . . . . 13 ⊢ 1o ⊆ suc 1o
17		df-2o 8298	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
18	16, 17	sseqtrri 3958	. . . . . . . . . . . 12 ⊢ 1o ⊆ 2o
19		1on 8309	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
20	17, 19	sucneqoni 35537	. . . . . . . . . . . . 13 ⊢ 2o ≠ 1o
21	20	necomi 2998	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
22		df-pss 3906	. . . . . . . . . . . 12 ⊢ (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
23	18, 21, 22	mpbir2an 708	. . . . . . . . . . 11 ⊢ 1o ⊊ 2o
24		ssnpss 4038	. . . . . . . . . . 11 ⊢ (2o ⊆ 1o → ¬ 1o ⊊ 2o)
25	23, 24	mt2 199	. . . . . . . . . 10 ⊢ ¬ 2o ⊆ 1o
26		sseq2 3947	. . . . . . . . . 10 ⊢ (𝑁 = 1o → (2o ⊆ 𝑁 ↔ 2o ⊆ 1o))
27	25, 26	mtbiri 327	. . . . . . . . 9 ⊢ (𝑁 = 1o → ¬ 2o ⊆ 𝑁)
28	27	con2i 139	. . . . . . . 8 ⊢ (2o ⊆ 𝑁 → ¬ 𝑁 = 1o)
29	28	intnanrd 490	. . . . . . 7 ⊢ (2o ⊆ 𝑁 → ¬ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈))
30	29	iffalsed 4470	. . . . . 6 ⊢ (2o ⊆ 𝑁 → if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
31		iftrue 4465	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
32	30, 31	sylan9eq 2798	. . . . 5 ⊢ ((2o ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
33	15, 32	sylan9eqr 2800	. . . 4 ⊢ (((2o ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
34	33	adantlll 715	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
35		simpll 764	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36		elex 3450	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
37	36	adantl 482	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38		opex 5379	. . . 4 ⊢ ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V
39	38	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V)
40	2, 34, 35, 37, 39	ovmpod 7425	. 2 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
41		df-ov 7278	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
42	40, 41	eqtr3di 2793	1 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432   ⊆ wss 3887   ⊊ wpss 3888  ∅c0 4256  ifcif 4459  ⟨cop 4567  ∪ cuni 4839   × cxp 5587  suc csuc 6268  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712  1^st c1st 7829  1_oc1o 8290  2_oc2o 8291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298

This theorem is referenced by:  finxpreclem4  35565