Theorem catprsc2 47722

Description: An alternate construction of the preorder induced by a category. See catprs2 47720 for details. See also catprsc 47721 for a different construction. The two constructions are different because df-cat 17617 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 5159	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}𝑤))
3		vex 3477	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3477	. . . . 5 ⊢ 𝑤 ∈ V
5		oveq12 7421	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
6	5	neeq1d 2999	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
7		eqid 2731	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}
8	3, 4, 6, 7	braba 5537	. . . 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 3199	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∅c0 4322   class class class wbr 5148  {copab 5210  (class class class)co 7412

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7415

This theorem is referenced by: (None)