Theorem catprsc 47720

Description: A construction of the preorder induced by a category. See catprs2 47719 for details. See also catprsc2 47721 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 5158	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤))
3		vex 3476	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3476	. . . . 5 ⊢ 𝑤 ∈ V
5		simpl 481	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
6	5	eleq1d 2816	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
7		simpr 483	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
8	7	eleq1d 2816	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
9		oveq12 7420	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
10	9	neeq1d 2998	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
11	6, 8, 10	3anbi123d 1434	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅)))
12		df-3an 1087	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
13	11, 12	bitrdi 286	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
14		eqid 2730	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}
15	3, 4, 13, 14	braba 5536	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
16	2, 15	bitrdi 286	. . 3 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
17	16	baibd 538	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
18	17	ralrimivva 3198	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∅c0 4321   class class class wbr 5147  {copab 5209  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6494  df-fv 6550  df-ov 7414

This theorem is referenced by: (None)