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Theorem finxpreclem3 33230

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

**Hypothesis**
Ref	Expression
finxpreclem3.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

**Assertion**
Ref	Expression
finxpreclem3	⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘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 6653	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
2		finxpreclem3.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3	2	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
4		eqeq1 2626	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1_𝑜 ↔ 𝑁 = 1_𝑜))
5		eleq1 2689	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
6	4, 5	bi2anan9 917	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈)))
7		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8	7	adantl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9		unieq 4444	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
11		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1^st ‘𝑥) = (1^st ‘𝑋))
12	11	adantl 482	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1^st ‘𝑥) = (1^st ‘𝑋))
13	10, 12	opeq12d 4410	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1^st ‘𝑥)⟩ = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
14		opeq12 4404	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
15	8, 13, 14	ifbieq12d 4113	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
16	6, 15	ifbieq2d 4111	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)))
17		sssucid 5802	. . . . . . . . . . . . 13 ⊢ 1_𝑜 ⊆ suc 1_𝑜
18		df-2o 7561	. . . . . . . . . . . . 13 ⊢ 2_𝑜 = suc 1_𝑜
19	17, 18	sseqtr4i 3638	. . . . . . . . . . . 12 ⊢ 1_𝑜 ⊆ 2_𝑜
20		1on 7567	. . . . . . . . . . . . . 14 ⊢ 1_𝑜 ∈ On
21	18, 20	sucneqoni 33214	. . . . . . . . . . . . 13 ⊢ 2_𝑜 ≠ 1_𝑜
22	21	necomi 2848	. . . . . . . . . . . 12 ⊢ 1_𝑜 ≠ 2_𝑜
23		df-pss 3590	. . . . . . . . . . . 12 ⊢ (1_𝑜 ⊊ 2_𝑜 ↔ (1_𝑜 ⊆ 2_𝑜 ∧ 1_𝑜 ≠ 2_𝑜))
24	19, 22, 23	mpbir2an 955	. . . . . . . . . . 11 ⊢ 1_𝑜 ⊊ 2_𝑜
25		ssnpss 3710	. . . . . . . . . . 11 ⊢ (2_𝑜 ⊆ 1_𝑜 → ¬ 1_𝑜 ⊊ 2_𝑜)
26	24, 25	mt2 191	. . . . . . . . . 10 ⊢ ¬ 2_𝑜 ⊆ 1_𝑜
27		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑁 = 1_𝑜 → (2_𝑜 ⊆ 𝑁 ↔ 2_𝑜 ⊆ 1_𝑜))
28	26, 27	mtbiri 317	. . . . . . . . 9 ⊢ (𝑁 = 1_𝑜 → ¬ 2_𝑜 ⊆ 𝑁)
29	28	con2i 134	. . . . . . . 8 ⊢ (2_𝑜 ⊆ 𝑁 → ¬ 𝑁 = 1_𝑜)
30	29	intnanrd 963	. . . . . . 7 ⊢ (2_𝑜 ⊆ 𝑁 → ¬ (𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈))
31	30	iffalsed 4097	. . . . . 6 ⊢ (2_𝑜 ⊆ 𝑁 → if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
32		iftrue 4092	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
33	31, 32	sylan9eq 2676	. . . . 5 ⊢ ((2_𝑜 ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
34	16, 33	sylan9eqr 2678	. . . 4 ⊢ (((2_𝑜 ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
35	34	adantlll 754	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
36		simpll 790	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37		elex 3212	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
38	37	adantl 482	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39		opex 4932	. . . 4 ⊢ ⟨∪ 𝑁, (1^st ‘𝑋)⟩ ∈ V
40	39	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ ∈ V)
41	3, 35, 36, 38, 40	ovmpt2d 6788	. 2 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
42	1, 41	syl5reqr 2671	1 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ⊊ wpss 3575 ∅c0 3915 ifcif 4086 ⟨cop 4183 ∪ cuni 4436 × cxp 5112 suc csuc 5725 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ωcom 7065 1^st c1st 7166 1_𝑜c1o 7553 2_𝑜c2o 7554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1o 7560 df-2o 7561

This theorem is referenced by: finxpreclem4 33231

W3C validator