Theorem lrrecval 28358

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 2852	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦)) ↔ 𝐴 ∈ ((L‘𝑦) ∪ (R‘𝑦))))
2		fveq2 6911	. . . 4 ⊢ (𝑦 = 𝐵 → (L‘𝑦) = (L‘𝐵))
3		fveq2 6911	. . . 4 ⊢ (𝑦 = 𝐵 → (R‘𝑦) = (R‘𝐵))
4	2, 3	uneq12d 4171	. . 3 ⊢ (𝑦 = 𝐵 → ((L‘𝑦) ∪ (R‘𝑦)) = ((L‘𝐵) ∪ (R‘𝐵)))
5	4	eleq2d 2850	. 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 5519	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ ((L‘𝐵) ∪ (R‘𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∪ cun 4152   class class class wbr 5106  {copab 5171  ‘cfv 6561  Nocsur 28030  Lcleft 28244  Rcright 28245

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-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-iota 6515  df-fv 6569

This theorem is used by:  lrrecval2  28359  lrrecse  28361  lrrecpred  28363