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Theorem isthincd 48837
Description: The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthincd.b (𝜑𝐵 = (Base‘𝐶))
isthincd.h (𝜑𝐻 = (Hom ‘𝐶))
isthincd.t ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
isthincd.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
isthincd (𝜑𝐶 ∈ ThinCat)
Distinct variable groups:   𝑦,𝐵   𝐶,𝑓,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝐻(𝑥,𝑦,𝑓)

Proof of Theorem isthincd
StepHypRef Expression
1 isthincd.c . 2 (𝜑𝐶 ∈ Cat)
2 isthincd.t . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
32ralrimivva 3200 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
4 isthincd.b . . . 4 (𝜑𝐵 = (Base‘𝐶))
5 isthincd.h . . . . . . . 8 (𝜑𝐻 = (Hom ‘𝐶))
65oveqd 7448 . . . . . . 7 (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
76eleq2d 2825 . . . . . 6 (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
87mobidv 2547 . . . . 5 (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
94, 8raleqbidv 3344 . . . 4 (𝜑 → (∀𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
104, 9raleqbidv 3344 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
113, 10mpbid 232 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
12 eqid 2735 . . 3 (Base‘𝐶) = (Base‘𝐶)
13 eqid 2735 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
1412, 13isthinc 48821 . 2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
151, 11, 14sylanbrc 583 1 (𝜑𝐶 ∈ ThinCat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ∃*wmo 2536  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  ThinCatcthinc 48819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-thinc 48820
This theorem is referenced by:  isthincd2  48838  oppcthin  48839  subthinc  48840  setcthin  48856
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