Theorem thincpropd 50220

Description: Two structures with the same base, hom-sets and composition operation are either both thin categories or neither. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
thincpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
thincpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
thincpropd.3	⊢ (𝜑 → 𝐶 ∈ 𝑉)
thincpropd.4	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
thincpropd	⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))

Proof of Theorem thincpropd

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		thincpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2		thincpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
3		thincpropd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		thincpropd.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5	1, 2, 3, 4	catpropd 17760	. . 3 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
6		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
10		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
11		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
12	6, 7, 8, 9, 10, 11	homfeqval 17748	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
13	12	eleq2d 2849	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
14	13	mobidv 2577	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
15	14	2ralbidva 3227	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
16	1	homfeqbas 17747	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
17	16	raleqdv 3323	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
18	16, 17	raleqbidv 3338	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
19	15, 18	bitrd 282	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
20	5, 19	anbi12d 643	. 2 ⊢ (𝜑 → ((𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ↔ (𝐷 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))))
21	6, 7	isthinc 50197	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
22		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
23	22, 8	isthinc 50197	. 2 ⊢ (𝐷 ∈ ThinCat ↔ (𝐷 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
24	20, 21, 23	3bitr4g 317	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  ∀wral 3079  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  Catccat 17715  Hom_f chomf 17717  comp_fccomf 17718  ThinCatcthinc 50195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-cat 17719  df-homf 17721  df-comf 17722  df-thinc 50196

This theorem is referenced by:  termcpropd  50281