Theorem rp-brsslt 42902

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 27746. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.)

Hypothesis

Ref	Expression
nla0001.defsslt	⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}

Assertion

Ref	Expression
rp-brsslt	⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		sseq1 4007	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑆))
2		raleq 3320	. . 3 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦))
3	1, 2	3anbi13d 1434	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦) ↔ (𝐴 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)))
4		sseq1 4007	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑆 ↔ 𝐵 ⊆ 𝑆))
5		raleq 3320	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))
6	5	ralbidv 3175	. . 3 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))
7	4, 6	3anbi23d 1435	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦) ↔ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)))
8		nla0001.defsslt	. 2 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
9	3, 7, 8	bropabg 42801	1 ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  Vcvv 3473   ⊆ wss 3949   class class class wbr 5152  {copab 5214

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-ral 3059  df-rex 3068  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-br 5153  df-opab 5215  df-xp 5688

This theorem is referenced by:  nla0002  42903  nla0003  42904