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Theorem isthinc 47641
Description: The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthinc.b 𝐵 = (Base‘𝐶)
isthinc.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
isthinc (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Distinct variable groups:   𝐵,𝑓,𝑥,𝑦   𝐶,𝑓,𝑥,𝑦   𝑓,𝐻,𝑥,𝑦

Proof of Theorem isthinc
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6907 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2 fveq2 6892 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 isthinc.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2791 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fvexd 6907 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6 fveq2 6892 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 isthinc.h . . . . . 6 𝐻 = (Hom ‘𝐶)
86, 7eqtr4di 2791 . . . . 5 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98adantr 482 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10 raleq 3323 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
1110raleqbi1dv 3334 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
1211ad2antlr 726 . . . . 5 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (∀𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
13 oveq 7415 . . . . . . . . 9 ( = 𝐻 → (𝑥𝑦) = (𝑥𝐻𝑦))
1413eleq2d 2820 . . . . . . . 8 ( = 𝐻 → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1514mobidv 2544 . . . . . . 7 ( = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16152ralbidv 3219 . . . . . 6 ( = 𝐻 → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
1716adantl 483 . . . . 5 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
1812, 17bitrd 279 . . . 4 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (∀𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
195, 9, 18sbcied2 3825 . . 3 ((𝑐 = 𝐶𝑏 = 𝐵) → ([(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
201, 4, 19sbcied2 3825 . 2 (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
21 df-thinc 47640 . 2 ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦)}
2220, 21elrab2 3687 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  ∃*wmo 2533  wral 3062  Vcvv 3475  [wsbc 3778  cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  Catccat 17608  ThinCatcthinc 47639
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-thinc 47640
This theorem is referenced by:  isthinc2  47642  isthinc3  47643  thincc  47644  thincmo2  47648  thincmoALT  47650  isthincd  47657  0thincg  47670
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