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Theorem rp-brsslt 42663
Description: Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27634. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.)
Hypothesis
Ref Expression
nla0001.defsslt < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
Assertion
Ref Expression
rp-brsslt (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴𝑆𝐵𝑆 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦   𝐵,𝑎,𝑏,𝑥,𝑦   𝑅,𝑎,𝑏   𝑆,𝑎,𝑏
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   < (𝑥,𝑦,𝑎,𝑏)

Proof of Theorem rp-brsslt
StepHypRef Expression
1 sseq1 3999 . . 3 (𝑎 = 𝐴 → (𝑎𝑆𝐴𝑆))
2 raleq 3314 . . 3 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦))
31, 23anbi13d 1434 . 2 (𝑎 = 𝐴 → ((𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦) ↔ (𝐴𝑆𝑏𝑆 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦)))
4 sseq1 3999 . . 3 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
5 raleq 3314 . . . 4 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 𝑥𝑅𝑦))
65ralbidv 3169 . . 3 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦))
74, 63anbi23d 1435 . 2 (𝑏 = 𝐵 → ((𝐴𝑆𝑏𝑆 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦) ↔ (𝐴𝑆𝐵𝑆 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦)))
8 nla0001.defsslt . 2 < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
93, 7, 8bropabg 42562 1 (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴𝑆𝐵𝑆 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  wss 3940   class class class wbr 5138  {copab 5200
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672
This theorem is referenced by:  nla0002  42664  nla0003  42665
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