Theorem lrrecval 28331

Description: The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
lrrec.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}

Assertion

Ref	Expression
lrrecval	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ ((L‘𝐵) ∪ (R‘𝐵))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem lrrecval

Step	Hyp	Ref	Expression
1		eleq1 2848	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦)) ↔ 𝐴 ∈ ((L‘𝑦) ∪ (R‘𝑦))))
2		fveq2 6886	. . . 4 ⊢ (𝑦 = 𝐵 → (L‘𝑦) = (L‘𝐵))
3		fveq2 6886	. . . 4 ⊢ (𝑦 = 𝐵 → (R‘𝑦) = (R‘𝐵))
4	2, 3	uneq12d 4168	. . 3 ⊢ (𝑦 = 𝐵 → ((L‘𝑦) ∪ (R‘𝑦)) = ((L‘𝐵) ∪ (R‘𝐵)))
5	4	eleq2d 2846	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ ((L‘𝑦) ∪ (R‘𝑦)) ↔ 𝐴 ∈ ((L‘𝐵) ∪ (R‘𝐵))))
6		lrrec.1	. 2 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}
7	1, 5, 6	brabg 5513	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ ((L‘𝐵) ∪ (R‘𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∪ cun 4149   class class class wbr 5102  {copab 5166  ‘cfv 6539  Nocsur 28003  Lcleft 28217  Rcright 28218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6547

This theorem is used by:  lrrecval2  28332  lrrecse  28334  lrrecpred  28336