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Theorem catprsc 45822
Description: A construction of the preorder induced by a category. See catprs2 45821 for details. See also catprsc2 45823 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypothesis
Ref Expression
catprsc.1 (𝜑 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})
Assertion
Ref Expression
catprsc (𝜑 → ∀𝑧𝐵𝑤𝐵 (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
StepHypRef Expression
1 catprsc.1 . . . . 5 (𝜑 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})
21breqd 5051 . . . 4 (𝜑 → (𝑧 𝑤𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤))
3 vex 3404 . . . . 5 𝑧 ∈ V
4 vex 3404 . . . . 5 𝑤 ∈ V
5 simpl 486 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
65eleq1d 2818 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
7 simpr 488 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
87eleq1d 2818 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐵))
9 oveq12 7191 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
109neeq1d 2994 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
116, 8, 103anbi123d 1437 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧𝐵𝑤𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅)))
12 df-3an 1090 . . . . . 6 ((𝑧𝐵𝑤𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅) ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
1311, 12bitrdi 290 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
14 eqid 2739 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}
153, 4, 13, 14braba 5402 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤 ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
162, 15bitrdi 290 . . 3 (𝜑 → (𝑧 𝑤 ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
1716baibd 543 . 2 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
1817ralrimivva 3104 1 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935  wral 3054  c0 4221   class class class wbr 5040  {copab 5102  (class class class)co 7182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-iota 6307  df-fv 6357  df-ov 7185
This theorem is referenced by: (None)
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