on2recsfn - Mathbox for Scott Fenton

	Mathbox for Scott Fenton		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > on2recsfn		Structured version Visualization version GIF version

Theorem on2recsfn 33826

Description: Show that double recursion over ordinals yields a function over pairs of ordinals. (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
on2recsfn	⊢ 𝐹 Fn (On × On)

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

**Proof of Theorem on2recsfn**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ {⟨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))}
2		onfr 6305	. . . . 5 ⊢ E Fr On
3	2	a1i 11	. . . 4 ⊢ (⊤ → E Fr On)
4	1, 3, 3	frxp2 33791	. . 3 ⊢ (⊤ → {⟨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))
5	4	mptru 1546	. 2 ⊢ {⟨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)
6		epweon 7625	. . . . . 6 ⊢ E We On
7		weso 5580	. . . . . 6 ⊢ ( E We On → E Or On)
8		sopo 5522	. . . . . 6 ⊢ ( E Or On → E Po On)
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ E Po On
10	9	a1i 11	. . . 4 ⊢ (⊤ → E Po On)
11	1, 10, 10	poxp2 33790	. . 3 ⊢ (⊤ → {⟨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))
12	11	mptru 1546	. 2 ⊢ {⟨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)
13		epse 5572	. . . . 5 ⊢ E Se On
14	13	a1i 11	. . . 4 ⊢ (⊤ → E Se On)
15	1, 14, 14	sexp2 33793	. . 3 ⊢ (⊤ → {⟨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))
16	15	mptru 1546	. 2 ⊢ {⟨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)
17		on2recs.1	. . 3 ⊢ 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)
18	17	fpr1 8119	. 2 ⊢ (({⟨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)) → 𝐹 Fn (On × On))
19	5, 12, 16, 18	mp3an 1460	1 ⊢ 𝐹 Fn (On × On)

Colors of variables: wff setvar class

Syntax hints: ∨ wo 844 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 {copab 5136 E cep 5494 Po wpo 5501 Or wor 5502 Fr wfr 5541 Se wse 5542 We wwe 5543 × cxp 5587 Oncon0 6266 Fn wfn 6428 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830 frecscfrecs 8096

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-1st 7831 df-2nd 7832 df-frecs 8097

This theorem is referenced by: naddfn 33830

W3C validator