Theorem catprsc 48802

Description: A construction of the preorder induced by a category. See catprs2 48801 for details. See also catprsc2 48803 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypothesis

Ref	Expression
catprsc.1	⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})

Assertion

Ref	Expression
catprsc	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

Distinct variable groups:   𝑤,𝐵,𝑥,𝑦   𝑥,𝐻,𝑦   𝜑,𝑤,𝑧   𝑥,𝑧,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑧)   𝐻(𝑧,𝑤)   ≤ (𝑥,𝑦,𝑧,𝑤)

Proof of Theorem catprsc

Step	Hyp	Ref	Expression
1		catprsc.1	. . . . 5 ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})
2	1	breqd 5159	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤))
3		vex 3482	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3482	. . . . 5 ⊢ 𝑤 ∈ V
5		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
6	5	eleq1d 2824	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
7		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
8	7	eleq1d 2824	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
9		oveq12 7440	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
10	9	neeq1d 2998	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
11	6, 8, 10	3anbi123d 1435	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅)))
12		df-3an 1088	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
13	11, 12	bitrdi 287	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
14		eqid 2735	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}
15	3, 4, 13, 14	braba 5547	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
16	2, 15	bitrdi 287	. . 3 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
17	16	baibd 539	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
18	17	ralrimivva 3200	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∅c0 4339   class class class wbr 5148  {copab 5210  (class class class)co 7431

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-ne 2939  df-ral 3060  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  df-ov 7434

This theorem is referenced by: (None)