Theorem thincpropd 49404

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 17646	. . 3 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
6		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
10		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
11		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
12	6, 7, 8, 9, 10, 11	homfeqval 17634	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
13	12	eleq2d 2814	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
14	13	mobidv 2542	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
15	14	2ralbidva 3197	. . . 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 17633	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
17	16	raleqdv 3296	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
18	16, 17	raleqbidv 3316	. . . 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 279	. . 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 632	. 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 49381	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
22		eqid 2729	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
23	22, 8	isthinc 49381	. 2 ⊢ (𝐷 ∈ ThinCat ↔ (𝐷 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
24	20, 21, 23	3bitr4g 314	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  ∀wral 3044  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604  ThinCatcthinc 49379

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-cat 17605  df-homf 17607  df-comf 17608  df-thinc 49380

This theorem is referenced by:  termcpropd  49465