Theorem thincpropd 49801

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 17644	. . 3 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
6		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
10		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
11		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
12	6, 7, 8, 9, 10, 11	homfeqval 17632	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
13	12	eleq2d 2823	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
14	13	mobidv 2550	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
15	14	2ralbidva 3200	. . . 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 17631	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
17	16	raleqdv 3298	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐷)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)))
18	16, 17	raleqbidv 3318	. . . 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 633	. 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 49778	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
22		eqid 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
23	22, 8	isthinc 49778	. 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 1542   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  Hom_f chomf 17601  comp_fccomf 17602  ThinCatcthinc 49776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-cat 17603  df-homf 17605  df-comf 17606  df-thinc 49777

This theorem is referenced by:  termcpropd  49862