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Theorem catprsc 46294
Description: A construction of the preorder induced by a category. See catprs2 46293 for details. See also catprsc2 46295 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 5085 . . . 4 (𝜑 → (𝑧 𝑤𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤))
3 vex 3436 . . . . 5 𝑧 ∈ V
4 vex 3436 . . . . 5 𝑤 ∈ V
5 simpl 483 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
65eleq1d 2823 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐵𝑧𝐵))
7 simpr 485 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
87eleq1d 2823 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐵))
9 oveq12 7284 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐻𝑦) = (𝑧𝐻𝑤))
109neeq1d 3003 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
116, 8, 103anbi123d 1435 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧𝐵𝑤𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅)))
12 df-3an 1088 . . . . . 6 ((𝑧𝐵𝑤𝐵 ∧ (𝑧𝐻𝑤) ≠ ∅) ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
1311, 12bitrdi 287 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅) ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
14 eqid 2738 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}
153, 4, 13, 14braba 5450 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}𝑤 ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅))
162, 15bitrdi 287 . . 3 (𝜑 → (𝑧 𝑤 ↔ ((𝑧𝐵𝑤𝐵) ∧ (𝑧𝐻𝑤) ≠ ∅)))
1716baibd 540 . 2 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
1817ralrimivva 3123 1 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064  c0 4256   class class class wbr 5074  {copab 5136  (class class class)co 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-ov 7278
This theorem is referenced by: (None)
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