Theorem finxpreclem3 34810

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		df-ov 7138	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
2		finxpreclem3.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3	2	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⟩))))
4		eqeq1 2802	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1o ↔ 𝑁 = 1o))
5		eleq1 2877	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
6	4, 5	bi2anan9 638	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈)))
7		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8	7	adantl 485	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9		unieq 4811	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
10	9	adantr 484	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
11		fveq2 6645	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1st ‘𝑥) = (1st ‘𝑋))
12	11	adantl 485	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1st ‘𝑥) = (1st ‘𝑋))
13	10, 12	opeq12d 4773	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
14		opeq12 4767	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
15	8, 13, 14	ifbieq12d 4452	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
16	6, 15	ifbieq2d 4450	. . . . 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⟩)))
17		sssucid 6236	. . . . . . . . . . . . 13 ⊢ 1o ⊆ suc 1o
18		df-2o 8086	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
19	17, 18	sseqtrri 3952	. . . . . . . . . . . 12 ⊢ 1o ⊆ 2o
20		1on 8092	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
21	18, 20	sucneqoni 34783	. . . . . . . . . . . . 13 ⊢ 2o ≠ 1o
22	21	necomi 3041	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
23		df-pss 3900	. . . . . . . . . . . 12 ⊢ (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
24	19, 22, 23	mpbir2an 710	. . . . . . . . . . 11 ⊢ 1o ⊊ 2o
25		ssnpss 4031	. . . . . . . . . . 11 ⊢ (2o ⊆ 1o → ¬ 1o ⊊ 2o)
26	24, 25	mt2 203	. . . . . . . . . 10 ⊢ ¬ 2o ⊆ 1o
27		sseq2 3941	. . . . . . . . . 10 ⊢ (𝑁 = 1o → (2o ⊆ 𝑁 ↔ 2o ⊆ 1o))
28	26, 27	mtbiri 330	. . . . . . . . 9 ⊢ (𝑁 = 1o → ¬ 2o ⊆ 𝑁)
29	28	con2i 141	. . . . . . . 8 ⊢ (2o ⊆ 𝑁 → ¬ 𝑁 = 1o)
30	29	intnanrd 493	. . . . . . 7 ⊢ (2o ⊆ 𝑁 → ¬ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈))
31	30	iffalsed 4436	. . . . . 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⟩))
32		iftrue 4431	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
33	31, 32	sylan9eq 2853	. . . . 5 ⊢ ((2o ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
34	16, 33	sylan9eqr 2855	. . . 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)⟩)
35	34	adantlll 717	. . 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)⟩)
36		simpll 766	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37		elex 3459	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
38	37	adantl 485	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39		opex 5321	. . . 4 ⊢ ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V
40	39	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V)
41	3, 35, 36, 38, 40	ovmpod 7281	. 2 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
42	1, 41	syl5reqr 2848	1 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ⊆ wss 3881   ⊊ wpss 3882  ∅c0 4243  ifcif 4425  ⟨cop 4531  ∪ cuni 4800   × cxp 5517  suc csuc 6161  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7560  1^st c1st 7669  1_oc1o 8078  2_oc2o 8079

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-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  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-rab 3115  df-v 3443  df-sbc 3721  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-br 5031  df-opab 5093  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-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1o 8085  df-2o 8086

This theorem is referenced by:  finxpreclem4  34811