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Theorem lrrecval 27987
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 2827 . 2 (𝑥 = 𝐴 → (𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)) ↔ 𝐴 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))))
2 fveq2 6907 . . . 4 (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵))
3 fveq2 6907 . . . 4 (𝑦 = 𝐵 → ( R ‘𝑦) = ( R ‘𝐵))
42, 3uneq12d 4179 . . 3 (𝑦 = 𝐵 → (( L ‘𝑦) ∪ ( R ‘𝑦)) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
54eleq2d 2825 . 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 5549 1 ((𝐴 No 𝐵 No ) → (𝐴𝑅𝐵𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cun 3961   class class class wbr 5148  {copab 5210  cfv 6563   No csur 27699   L cleft 27899   R cright 27900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571
This theorem is referenced by:  lrrecval2  27988  lrrecse  27990  lrrecpred  27992
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