Theorem norec2fn 28369

Description: The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)

Hypothesis

Ref	Expression
norec2.1	⊢ 𝐹 = norec2 (𝐺)

Assertion

Ref	Expression
norec2fn	⊢ 𝐹 Fn (No × No)

Proof of Theorem norec2fn

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))}
2		eqid 2763	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}
3	1, 2	noxpordfr 28364	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr (No × No)
4	1, 2	noxpordpo 28363	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po (No × No)
5	1, 2	noxpordse 28365	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se (No × No)
6		norec2.1	. . . 4 ⊢ 𝐹 = norec2 (𝐺)
7		df-norec2 28362	. . . 4 ⊢ norec2 (𝐺) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), 𝐺)
8	6, 7	eqtri 2786	. . 3 ⊢ 𝐹 = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), 𝐺)
9	8	fpr1 8346	. 2 ⊢ (({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se (No × No)) → 𝐹 Fn (No × No))
10	3, 4, 5, 9	mp3an 1491	1 ⊢ 𝐹 Fn (No × No)

This proof depends on syntax axioms:   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∪ cun 4152   class class class wbr 5106  {copab 5170   Po wpo 5561   Fr wfr 5605   Se wse 5606   × cxp 5653   Fn wfn 6550  ‘cfv 6555  1^st c1st 8020  2^nd c2nd 8021  frecscfrecs 8323  Nocsur 28024  Lcleft 28238  Rcright 28239   norec2 cnorec2 28361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec2 28362

This theorem is used by:  addsfn  28374  mulsfn  28521