Theorem catprsc 49819

Description: A construction of the preorder induced by a category. See catprs2 49818 for details. See also catprsc2 49820 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 5120	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤))
3		vex 3459	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3459	. . . . 5 ⊢ 𝑤 ∈ V
5		simpl 487	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
6	5	eleq1d 2848	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
7		simpr 489	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
8	7	eleq1d 2848	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
9		oveq12 7419	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
10	9	neeq1d 3017	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
11	6, 8, 10	3anbi123d 1464	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅)))
12		df-3an 1105	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
13	11, 12	bitrdi 290	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
14		eqid 2763	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}
15	3, 4, 13, 14	braba 5521	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
16	2, 15	bitrdi 290	. . 3 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
17	16	baibd 548	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
18	17	ralrimivva 3208	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∅c0 4286   class class class wbr 5109  {copab 5173  (class class class)co 7410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is used by: (None)