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Theorem on2recsov 8655

Description: Calculate the value of the double ordinal recursion operator. (Contributed by Scott Fenton, 3-Sep-2024.)

**Hypothesis**
Ref	Expression
on2recs.1	⊢ 𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)

**Assertion**
Ref	Expression
on2recsov	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐵(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦)

**Proof of Theorem on2recsov**
Step	Hyp	Ref	Expression
1		df-ov 7415	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		opelxp 5699	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (On × On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On))
3		eqid 2763	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
4		onfr 6402	. . . . . . . . 9 ⊢ E Fr On
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Fr On)
6	3, 5, 5	frxp2 8141	. . . . . . 7 ⊢ (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Fr (On × On))
7	6	mptru 1577	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Fr (On × On)
8		epweon 7775	. . . . . . . . . 10 ⊢ E We On
9		weso 5654	. . . . . . . . . 10 ⊢ ( E We On → E Or On)
10		sopo 5590	. . . . . . . . . 10 ⊢ ( E Or On → E Po On)
11	8, 9, 10	mp2b 10	. . . . . . . . 9 ⊢ E Po On
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Po On)
13	3, 12, 12	poxp2 8140	. . . . . . 7 ⊢ (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Po (On × On))
14	13	mptru 1577	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Po (On × On)
15		epse 5645	. . . . . . . . 9 ⊢ E Se On
16	15	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Se On)
17	3, 16, 16	sexp2 8143	. . . . . . 7 ⊢ (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Se (On × On))
18	17	mptru 1577	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Se (On × On)
19	7, 14, 18	3pm3.2i 1358	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Fr (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Po (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Se (On × On))
20		on2recs.1	. . . . . 6 ⊢ 𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)
21	20	fpr2 8302	. . . . 5 ⊢ ((({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Fr (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Po (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} Se (On × On)) ∧ ⟨𝐴, 𝐵⟩ ∈ (On × On)) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩))))
22	19, 21	mpan 702	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (On × On) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩))))
23	2, 22	sylbir 238	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩))))
24	1, 23	eqtrid 2810	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩))))
25	3	xpord2pred 8142	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩) = (((Pred( E , On, 𝐴) ∪ {𝐴}) × (Pred( E , On, 𝐵) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
26		predon 7786	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
27	26	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred( E , On, 𝐴) = 𝐴)
28	27	uneq1d 4122	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = (𝐴 ∪ {𝐴}))
29		df-suc 6368	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
30	28, 29	eqtr4di 2816	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = suc 𝐴)
31		predon 7786	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Pred( E , On, 𝐵) = 𝐵)
32	31	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred( E , On, 𝐵) = 𝐵)
33	32	uneq1d 4122	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = (𝐵 ∪ {𝐵}))
34		df-suc 6368	. . . . . . . 8 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
35	33, 34	eqtr4di 2816	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = suc 𝐵)
36	30, 35	xpeq12d 5694	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((Pred( E , On, 𝐴) ∪ {𝐴}) × (Pred( E , On, 𝐵) ∪ {𝐵})) = (suc 𝐴 × suc 𝐵))
37	36	difeq1d 4081	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((Pred( E , On, 𝐴) ∪ {𝐴}) × (Pred( E , On, 𝐵) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) = ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))
38	25, 37	eqtrd 2798	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩) = ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))
39	38	reseq2d 5980	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩)) = (𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩})))
40	39	oveq2d 7428	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), ⟨𝐴, 𝐵⟩))) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))
41	24, 40	eqtrd 2798	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∪ cun 3904 {csn 4590 ⟨cop 4596 class class class wbr 5110 {copab 5174 E cep 5562 Po wpo 5569 Or wor 5570 Fr wfr 5613 Se wse 5614 We wwe 5615 × cxp 5661 ↾ cres 5665 Predcpred 6303 Oncon0 6362 suc csuc 6364 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 frecscfrecs 8278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-1st 7987 df-2nd 7988 df-frecs 8279

This theorem is referenced by: naddcllem 8663 nmulprop 36660

W3C validator