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Theorem on2recsfn 8705
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) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (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
StepHypRef Expression
1 eqid 2737 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
2 onfr 6423 . . . . 5 E Fr On
32a1i 11 . . . 4 (⊤ → E Fr On)
41, 3, 3frxp2 8169 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Fr (On × On))
54mptru 1547 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Fr (On × On)
6 epweon 7795 . . . . . 6 E We On
7 weso 5676 . . . . . 6 ( E We On → E Or On)
8 sopo 5611 . . . . . 6 ( E Or On → E Po On)
96, 7, 8mp2b 10 . . . . 5 E Po On
109a1i 11 . . . 4 (⊤ → E Po On)
111, 10, 10poxp2 8168 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Po (On × On))
1211mptru 1547 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Po (On × On)
13 epse 5667 . . . . 5 E Se On
1413a1i 11 . . . 4 (⊤ → E Se On)
151, 14, 14sexp2 8171 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Se (On × On))
1615mptru 1547 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Se (On × On)
17 on2recs.1 . . 3 𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (On × On), 𝐺)
1817fpr1 8328 . 2 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Fr (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Po (On × On) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Se (On × On)) → 𝐹 Fn (On × On))
195, 12, 16, 18mp3an 1463 1 𝐹 Fn (On × On)
Colors of variables: wff setvar class
Syntax hints:  wo 848  w3a 1087   = wceq 1540  wtru 1541  wcel 2108  wne 2940   class class class wbr 5143  {copab 5205   E cep 5583   Po wpo 5590   Or wor 5591   Fr wfr 5634   Se wse 5635   We wwe 5636   × cxp 5683  Oncon0 6384   Fn wfn 6556  cfv 6561  1st c1st 8012  2nd c2nd 8013  frecscfrecs 8305
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-frecs 8306
This theorem is referenced by:  naddfn  8713
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