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Theorem on2recsov 8635

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 7393	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		opelxp 5677	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (On × On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On))
3		eqid 2730	. . . . . . . 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 6374	. . . . . . . . 9 ⊢ E Fr On
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Fr On)
6	3, 5, 5	frxp2 8126	. . . . . . 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 1547	. . . . . 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 7754	. . . . . . . . . 10 ⊢ E We On
9		weso 5632	. . . . . . . . . 10 ⊢ ( E We On → E Or On)
10		sopo 5568	. . . . . . . . . 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 8125	. . . . . . 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 1547	. . . . . 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 5623	. . . . . . . . 9 ⊢ E Se On
16	15	a1i 11	. . . . . . . 8 ⊢ (⊤ → E Se On)
17	3, 16, 16	sexp2 8128	. . . . . . 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 1547	. . . . . 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 1340	. . . . 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 8286	. . . . 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 690	. . . 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 2777	. 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 8127	. . . . 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 7765	. . . . . . . . . 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 4133	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = (𝐴 ∪ {𝐴}))
29		df-suc 6341	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
30	28, 29	eqtr4di 2783	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐴) ∪ {𝐴}) = suc 𝐴)
31		predon 7765	. . . . . . . . . 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 4133	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = (𝐵 ∪ {𝐵}))
34		df-suc 6341	. . . . . . . 8 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
35	33, 34	eqtr4di 2783	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Pred( E , On, 𝐵) ∪ {𝐵}) = suc 𝐵)
36	30, 35	xpeq12d 5672	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((Pred( E , On, 𝐴) ∪ {𝐴}) × (Pred( E , On, 𝐵) ∪ {𝐵})) = (suc 𝐴 × suc 𝐵))
37	36	difeq1d 4091	. . . . 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 2765	. . . 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 5953	. . 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 7406	. 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 2765	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 847 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 ∪ cun 3915 {csn 4592 ⟨cop 4598 class class class wbr 5110 {copab 5172 E cep 5540 Po wpo 5547 Or wor 5548 Fr wfr 5591 Se wse 5592 We wwe 5593 × cxp 5639 ↾ cres 5643 Predcpred 6276 Oncon0 6335 suc csuc 6337 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 frecscfrecs 8262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-1st 7971 df-2nd 7972 df-frecs 8263

This theorem is referenced by: naddcllem 8643

W3C validator