Theorem thincpropd 50519

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 17876	. . 3 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
6		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
10		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
11		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
12	6, 7, 8, 9, 10, 11	homfeqval 17864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
13	12	eleq2d 2847	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
14	13	mobidv 2575	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
15	14	2ralbidva 3225	. . . 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 17863	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
17	16	raleqdv 3320	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
18	16, 17	raleqbidv 3335	. . . 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 644	. 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 50496	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
22		eqid 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
23	22, 8	isthinc 50496	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃*wmo 2563  ∀wral 3077  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  Hom chom 17432  Catccat 17831  Hom_f chomf 17833  comp_fccomf 17834  ThinCatcthinc 50494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-cat 17835  df-homf 17837  df-comf 17838  df-thinc 50495

This theorem is used by:  termcpropd  50580