Theorem finxpreclem5 37889

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 7399	. . 3 ⊢ (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
3		0ex 5257	. . . . . . 7 ⊢ ∅ ∈ V
4		opex 5431	. . . . . . . 8 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
5		opex 5431	. . . . . . . 8 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
6	4, 5	ifex 4531	. . . . . . 7 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
7	3, 6	ifex 4531	. . . . . 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 7543	. . . . . 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 1474	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11	10	ad2antrr 736	. . . 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 8450	. . . . . . . . . . 11 ⊢ 1o ∈ On
13	12	onirri 6460	. . . . . . . . . 10 ⊢ ¬ 1o ∈ 1o
14		eleq2 2851	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (1o ∈ 𝑛 ↔ 1o ∈ 1o))
15	13, 14	mtbiri 329	. . . . . . . . 9 ⊢ (𝑛 = 1o → ¬ 1o ∈ 𝑛)
16	15	con2i 139	. . . . . . . 8 ⊢ (1o ∈ 𝑛 → ¬ 𝑛 = 1o)
17	16	intnanrd 493	. . . . . . 7 ⊢ (1o ∈ 𝑛 → ¬ (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))
18	17	iffalsed 4491	. . . . . 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 485	. . . . 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 4489	. . . . 5 ⊢ (¬ 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
21	19, 20	sylan9eq 2817	. . . 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 2797	. . 3 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
23	1, 22	eqtr3id 2811	. 2 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
24	23	ex 416	1 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ifcif 4480  ⟨cop 4588  ∪ cuni 4865   × cxp 5645  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  ωcom 7846  1^st c1st 7968  1_oc1o 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1o 8437

This theorem is referenced by:  finxpreclem6  37890