Theorem finxpreclem5 35493

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 7258	. . 3 ⊢ (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
3		0ex 5226	. . . . . . 7 ⊢ ∅ ∈ V
4		opex 5373	. . . . . . . 8 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
5		opex 5373	. . . . . . . 8 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
6	4, 5	ifex 4506	. . . . . . 7 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
7	3, 6	ifex 4506	. . . . . 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 7398	. . . . . 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 1451	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11	10	ad2antrr 722	. . . 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 8274	. . . . . . . . . . 11 ⊢ 1o ∈ On
13	12	onirri 6358	. . . . . . . . . 10 ⊢ ¬ 1o ∈ 1o
14		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (1o ∈ 𝑛 ↔ 1o ∈ 1o))
15	13, 14	mtbiri 326	. . . . . . . . 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 4467	. . . . . 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 4465	. . . . 5 ⊢ (¬ 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
21	19, 20	sylan9eq 2799	. . . 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 2778	. . 3 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
23	1, 22	eqtr3id 2793	. 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 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ifcif 4456  ⟨cop 4564  ∪ cuni 4836   × cxp 5578  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  ωcom 7687  1^st c1st 7802  1_oc1o 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1o 8267

This theorem is referenced by:  finxpreclem6  35494