|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > lrrecval | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | 
| Ref | Expression | 
|---|---|
| lrrecval | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) | 
| 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)))) | 
| Colors of variables: wff setvar class | 
| 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 | 
| Copyright terms: Public domain | W3C validator |