Theorem catprsc2 49851

Description: An alternate construction of the preorder induced by a category. See catprs2 49849 for details. See also catprsc 49850 for a different construction. The two constructions are different because df-cat 17748 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 5122	. . . 4 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}𝑤))
3		vex 3461	. . . . 5 ⊢ 𝑧 ∈ V
4		vex 3461	. . . . 5 ⊢ 𝑤 ∈ V
5		oveq12 7428	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
6	5	neeq1d 3019	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
7		eqid 2765	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}
8	3, 4, 6, 7	braba 5523	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)
9	2, 8	bitrdi 290	. . 3 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
10	9	adantr 486	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
11	10	ralrimivva 3210	1 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∅c0 4286   class class class wbr 5111  {copab 5175  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by: (None)