Theorem lrrecval 34105

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 6771	. . . 4 ⊢ (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵))
3		fveq2 6771	. . . 4 ⊢ (𝑦 = 𝐵 → ( R ‘𝑦) = ( R ‘𝐵))
4	2, 3	uneq12d 4103	. . 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 5455	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ∪ cun 3890   class class class wbr 5079  {copab 5141  ‘cfv 6432   No csur 33852   L cleft 34038   R cright 34039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-iota 6390  df-fv 6440

This theorem is referenced by:  lrrecval2  34106  lrrecse  34108  lrrecpred  34110