Theorem finxpreclem3 35219

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 2743	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1o ↔ 𝑁 = 1o))
4		eleq1 2821	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
5	3, 4	bi2anan9 639	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈)))
6		eleq1 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
7	6	adantl 485	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8		unieq 4817	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
9	8	adantr 484	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
10		fveq2 6686	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1st ‘𝑥) = (1st ‘𝑋))
11	10	adantl 485	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1st ‘𝑥) = (1st ‘𝑋))
12	9, 11	opeq12d 4779	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
13		opeq12 4773	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
14	7, 12, 13	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⟩))
15	5, 14	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⟩)))
16		sssucid 6259	. . . . . . . . . . . . 13 ⊢ 1o ⊆ suc 1o
17		df-2o 8144	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
18	16, 17	sseqtrri 3924	. . . . . . . . . . . 12 ⊢ 1o ⊆ 2o
19		1on 8150	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
20	17, 19	sucneqoni 35192	. . . . . . . . . . . . 13 ⊢ 2o ≠ 1o
21	20	necomi 2989	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
22		df-pss 3872	. . . . . . . . . . . 12 ⊢ (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
23	18, 21, 22	mpbir2an 711	. . . . . . . . . . 11 ⊢ 1o ⊊ 2o
24		ssnpss 4004	. . . . . . . . . . 11 ⊢ (2o ⊆ 1o → ¬ 1o ⊊ 2o)
25	23, 24	mt2 203	. . . . . . . . . 10 ⊢ ¬ 2o ⊆ 1o
26		sseq2 3913	. . . . . . . . . 10 ⊢ (𝑁 = 1o → (2o ⊆ 𝑁 ↔ 2o ⊆ 1o))
27	25, 26	mtbiri 330	. . . . . . . . 9 ⊢ (𝑁 = 1o → ¬ 2o ⊆ 𝑁)
28	27	con2i 141	. . . . . . . 8 ⊢ (2o ⊆ 𝑁 → ¬ 𝑁 = 1o)
29	28	intnanrd 493	. . . . . . 7 ⊢ (2o ⊆ 𝑁 → ¬ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈))
30	29	iffalsed 4435	. . . . . 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 4430	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
32	30, 31	sylan9eq 2794	. . . . 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 2796	. . . 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 718	. . 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 767	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36		elex 3418	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
37	36	adantl 485	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38		opex 5332	. . . 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 7329	. 2 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
41		df-ov 7185	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
42	40, 41	eqtr3di 2789	1 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ≠ wne 2935  Vcvv 3400   ⊆ wss 3853   ⊊ wpss 3854  ∅c0 4221  ifcif 4424  ⟨cop 4532  ∪ cuni 4806   × cxp 5533  suc csuc 6184  ‘cfv 6349  (class class class)co 7182   ∈ cmpo 7184  ωcom 7611  1^st c1st 7724  1_oc1o 8136  2_oc2o 8137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1o 8143  df-2o 8144

This theorem is referenced by:  finxpreclem4  35220