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Theorem isthinc 48821
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 6922 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2 fveq2 6907 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 isthinc.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2793 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fvexd 6922 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6 fveq2 6907 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 isthinc.h . . . . . 6 𝐻 = (Hom ‘𝐶)
86, 7eqtr4di 2793 . . . . 5 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98adantr 480 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10 raleq 3321 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
1110raleqbi1dv 3336 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
1211ad2antlr 727 . . . . 5 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (∀𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
13 oveq 7437 . . . . . . . . 9 ( = 𝐻 → (𝑥𝑦) = (𝑥𝐻𝑦))
1413eleq2d 2825 . . . . . . . 8 ( = 𝐻 → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1514mobidv 2547 . . . . . . 7 ( = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16152ralbidv 3219 . . . . . 6 ( = 𝐻 → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
1716adantl 481 . . . . 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 3839 . . 3 ((𝑐 = 𝐶𝑏 = 𝐵) → ([(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
201, 4, 19sbcied2 3839 . 2 (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
21 df-thinc 48820 . 2 ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦)}
2220, 21elrab2 3698 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  ∃*wmo 2536  wral 3059  Vcvv 3478  [wsbc 3791  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:  isthinc2  48822  isthinc3  48823  thincc  48824  thincmo2  48828  thincmoALT  48830  isthincd  48837  0thincg  48851
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