Theorem rp-brsslt 43526

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 27725. (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 3955	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑆))
2		raleq 3289	. . 3 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦))
3	1, 2	3anbi13d 1440	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦) ↔ (𝐴 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)))
4		sseq1 3955	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝑆 ↔ 𝐵 ⊆ 𝑆))
5		raleq 3289	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))
6	5	ralbidv 3155	. . 3 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))
7	4, 6	3anbi23d 1441	. 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 43426	1 ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089  {copab 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620

This theorem is referenced by:  nla0002  43527  nla0003  43528