Theorem lrrecval 27973

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 2828	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)) ↔ 𝐴 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))))
2		fveq2 6905	. . . 4 ⊢ (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵))
3		fveq2 6905	. . . 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 2826	. 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 5543	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ∪ cun 3948   class class class wbr 5142  {copab 5204  ‘cfv 6560   No csur 27685   L cleft 27885   R cright 27886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568

This theorem is referenced by:  lrrecval2  27974  lrrecse  27976  lrrecpred  27978