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Theorem on2recsfn 8704
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 2735 . . . 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 6425 . . . . 5 E Fr On
32a1i 11 . . . 4 (⊤ → E Fr On)
41, 3, 3frxp2 8168 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Fr (On × On))
54mptru 1544 . 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 7794 . . . . . 6 E We On
7 weso 5680 . . . . . 6 ( E We On → E Or On)
8 sopo 5616 . . . . . 6 ( E Or On → E Po On)
96, 7, 8mp2b 10 . . . . 5 E Po On
109a1i 11 . . . 4 (⊤ → E Po On)
111, 10, 10poxp2 8167 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Po (On × On))
1211mptru 1544 . 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 5671 . . . . 5 E Se On
1413a1i 11 . . . 4 (⊤ → E Se On)
151, 14, 14sexp2 8170 . . 3 (⊤ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} Se (On × On))
1615mptru 1544 . 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 8327 . 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 1460 1 𝐹 Fn (On × On)
Colors of variables: wff setvar class
Syntax hints:  wo 847  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  wne 2938   class class class wbr 5148  {copab 5210   E cep 5588   Po wpo 5595   Or wor 5596   Fr wfr 5638   Se wse 5639   We wwe 5640   × cxp 5687  Oncon0 6386   Fn wfn 6558  cfv 6563  1st c1st 8011  2nd c2nd 8012  frecscfrecs 8304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-frecs 8305
This theorem is referenced by:  naddfn  8712
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