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Theorem rp-brsslt 43413
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 27845. (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 4021 . . 3 (𝑎 = 𝐴 → (𝑎𝑆𝐴𝑆))
2 raleq 3321 . . 3 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦))
31, 23anbi13d 1437 . 2 (𝑎 = 𝐴 → ((𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦) ↔ (𝐴𝑆𝑏𝑆 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦)))
4 sseq1 4021 . . 3 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
5 raleq 3321 . . . 4 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 𝑥𝑅𝑦))
65ralbidv 3176 . . 3 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦))
74, 63anbi23d 1438 . 2 (𝑏 = 𝐵 → ((𝐴𝑆𝑏𝑆 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑅𝑦) ↔ (𝐴𝑆𝐵𝑆 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦)))
8 nla0001.defsslt . 2 < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
93, 7, 8bropabg 43313 1 (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴𝑆𝐵𝑆 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148  {copab 5210
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-ral 3060  df-rex 3069  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-br 5149  df-opab 5211  df-xp 5695
This theorem is referenced by:  nla0002  43414  nla0003  43415
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