Theorem catprsc2 48931

Description: An alternate construction of the preorder induced by a category. See catprs2 48929 for details. See also catprsc 48930 for a different construction. The two constructions are different because df-cat 17635 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.)

Hypothesis

Ref	Expression
catprsc2.1	⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})

Assertion

Ref	Expression
catprsc2	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

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

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

Proof of Theorem catprsc2

Step	Hyp	Ref	Expression
1		catprsc2.1	. . . . 5 ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})
2	1	breqd 5126	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}𝑤))
3		vex 3459	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3459	. . . . 5 ⊢ 𝑤 ∈ V
5		oveq12 7403	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
6	5	neeq1d 2986	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
7		eqid 2730	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}
8	3, 4, 6, 7	braba 5505	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)
9	2, 8	bitrdi 287	. . 3 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
10	9	adantr 480	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
11	10	ralrimivva 3182	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  ∀wral 3046  ∅c0 4304   class class class wbr 5115  {copab 5177  (class class class)co 7394

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-iota 6472  df-fv 6527  df-ov 7397

This theorem is referenced by: (None)