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Theorem lrrecval 27876
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 ) → (𝐴𝑅𝐵𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem lrrecval
StepHypRef Expression
1 eleq1 2817 . 2 (𝑥 = 𝐴 → (𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)) ↔ 𝐴 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))))
2 fveq2 6902 . . . 4 (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵))
3 fveq2 6902 . . . 4 (𝑦 = 𝐵 → ( R ‘𝑦) = ( R ‘𝐵))
42, 3uneq12d 4165 . . 3 (𝑦 = 𝐵 → (( L ‘𝑦) ∪ ( R ‘𝑦)) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
54eleq2d 2815 . 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))}
71, 5, 6brabg 5545 1 ((𝐴 No 𝐵 No ) → (𝐴𝑅𝐵𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cun 3947   class class class wbr 5152  {copab 5214  cfv 6553   No csur 27593   L cleft 27792   R cright 27793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-iota 6505  df-fv 6561
This theorem is referenced by:  lrrecval2  27877  lrrecse  27879  lrrecpred  27881
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