Theorem finxpreclem5 37728

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem5.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem5	⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Distinct variable group:   𝑥,𝑛

Allowed substitution hints:   𝑈(𝑥,𝑛)   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem5

Step	Hyp	Ref	Expression
1		df-ov 7364	. . 3 ⊢ (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
3		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
4		opex 5412	. . . . . . . 8 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
5		opex 5412	. . . . . . . 8 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
6	4, 5	ifex 4518	. . . . . . 7 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
7	3, 6	ifex 4518	. . . . . 6 ⊢ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
8		finxpreclem5.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
9	8	ovmpt4g 7508	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑥 ∈ V ∧ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V) → (𝑛𝐹𝑥) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
10	2, 7, 9	mp3an23 1456	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11	10	ad2antrr 727	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12		1on 8411	. . . . . . . . . . 11 ⊢ 1o ∈ On
13	12	onirri 6432	. . . . . . . . . 10 ⊢ ¬ 1o ∈ 1o
14		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (1o ∈ 𝑛 ↔ 1o ∈ 1o))
15	13, 14	mtbiri 327	. . . . . . . . 9 ⊢ (𝑛 = 1o → ¬ 1o ∈ 𝑛)
16	15	con2i 139	. . . . . . . 8 ⊢ (1o ∈ 𝑛 → ¬ 𝑛 = 1o)
17	16	intnanrd 489	. . . . . . 7 ⊢ (1o ∈ 𝑛 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
18	17	iffalsed 4478	. . . . . 6 ⊢ (1o ∈ 𝑛 → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
19	18	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
20		iffalse 4476	. . . . 5 ⊢ (¬ 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
21	19, 20	sylan9eq 2792	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
22	11, 21	eqtrd 2772	. . 3 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
23	1, 22	eqtr3id 2786	. 2 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
24	23	ex 412	1 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ifcif 4467  ⟨cop 4574  ∪ cuni 4851   × cxp 5623  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  ωcom 7811  1^st c1st 7934  1_oc1o 8392

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1o 8399

This theorem is referenced by:  finxpreclem6  37729