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Theorem on2recsov 8724

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 7451	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		opelxp 5736	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (On × On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On))
3		eqid 2740	. . . . . . . 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 6434	. . . . . . . . 9 ⊢ E Fr On
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Fr On)
6	3, 5, 5	frxp2 8185	. . . . . . 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 1544	. . . . . 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 7810	. . . . . . . . . 10 ⊢ E We On
9		weso 5691	. . . . . . . . . 10 ⊢ ( E We On → E Or On)
10		sopo 5627	. . . . . . . . . 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 8184	. . . . . . 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 1544	. . . . . 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 5682	. . . . . . . . 9 ⊢ E Se On
16	15	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Se On)
17	3, 16, 16	sexp2 8187	. . . . . . 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 1544	. . . . . 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 1339	. . . . 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 8345	. . . . 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 689	. . . 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 235	. . 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 2792	. 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 8186	. . . . 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 7821	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred( E , On, 𝐴) = 𝐴)
28	27	uneq1d 4190	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = (𝐴 ∪ {𝐴}))
29		df-suc 6401	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
30	28, 29	eqtr4di 2798	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = suc 𝐴)
31		predon 7821	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Pred( E , On, 𝐵) = 𝐵)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Pred( E , On, 𝐵) = 𝐵)
33	32	uneq1d 4190	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = (𝐵 ∪ {𝐵}))
34		df-suc 6401	. . . . . . . 8 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
35	33, 34	eqtr4di 2798	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = suc 𝐵)
36	30, 35	xpeq12d 5731	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((Pred( E , On, 𝐴) ∪ {𝐴}) × (Pred( E , On, 𝐵) ∪ {𝐵})) = (suc 𝐴 × suc 𝐵))
37	36	difeq1d 4148	. . . . 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 2780	. . . 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 6009	. . 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 7464	. 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 2780	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 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ∪ cun 3974 {csn 4648 ⟨cop 4654 class class class wbr 5166 {copab 5228 E cep 5598 Po wpo 5605 Or wor 5606 Fr wfr 5649 Se wse 5650 We wwe 5651 × cxp 5698 ↾ cres 5702 Predcpred 6331 Oncon0 6395 suc csuc 6397 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 frecscfrecs 8321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-1st 8030 df-2nd 8031 df-frecs 8322

This theorem is referenced by: naddcllem 8732

W3C validator