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Theorem isthinc 49069
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 6921 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2 fveq2 6906 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 isthinc.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2795 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fvexd 6921 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6 fveq2 6906 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 isthinc.h . . . . . 6 𝐻 = (Hom ‘𝐶)
86, 7eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98adantr 480 . . . 4 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10 raleq 3323 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝑦)))
1110raleqbi1dv 3338 . . . . . 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 2827 . . . . . . . 8 ( = 𝐻 → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1514mobidv 2549 . . . . . . 7 ( = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16152ralbidv 3221 . . . . . 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 3833 . . 3 ((𝑐 = 𝐶𝑏 = 𝐵) → ([(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
201, 4, 19sbcied2 3833 . 2 (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
21 df-thinc 49068 . 2 ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ]𝑥𝑏𝑦𝑏 ∃*𝑓 𝑓 ∈ (𝑥𝑦)}
2220, 21elrab2 3695 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  Vcvv 3480  [wsbc 3788  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  ThinCatcthinc 49067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-thinc 49068
This theorem is referenced by:  isthinc2  49070  isthinc3  49071  thincc  49072  thincmo2  49076  thincmoALT  49078  isthincd  49085  thincpropd  49091  0thincg  49107
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