Theorem functhincfun 49444

Description: A functor to a thin category is determined entirely by the object part. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
functhincfun.d	⊢ (𝜑 → 𝐶 ∈ Cat)
functhincfun.e	⊢ (𝜑 → 𝐷 ∈ ThinCat)

Assertion

Ref	Expression
functhincfun	⊢ (𝜑 → Fun (𝐶 Func 𝐷))

Proof of Theorem functhincfun

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

Step	Hyp	Ref	Expression
1		relfunc 17769	. 2 ⊢ Rel (𝐶 Func 𝐷)
2		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓(𝐶 Func 𝐷)𝑔)
3		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
5		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		functhincfun.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝐶 ∈ Cat)
9		functhincfun.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝐷 ∈ ThinCat)
11	3, 4, 2	funcf1 17773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
12		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))
13		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑓(𝐶 Func 𝐷)𝑔)
14		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
15		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
16	3, 5, 6, 13, 14, 15	funcf2 17775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝑔𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))
17	16	f002 48848	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)) = ∅ → (𝑥(Hom ‘𝐶)𝑦) = ∅))
18	17	ralrimivva 3172	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)) = ∅ → (𝑥(Hom ‘𝐶)𝑦) = ∅))
19	3, 4, 5, 6, 8, 10, 11, 12, 18	functhinc 49443	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → (𝑓(𝐶 Func 𝐷)𝑔 ↔ 𝑔 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))))
20	2, 19	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑔 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))))
21		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓(𝐶 Func 𝐷)ℎ)
22	3, 4, 5, 6, 8, 10, 11, 12, 18	functhinc 49443	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → (𝑓(𝐶 Func 𝐷)ℎ ↔ ℎ = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))))
23	21, 22	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → ℎ = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))))
24	20, 23	eqtr4d 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑔 = ℎ)
25	24	ex 412	. . . 4 ⊢ (𝜑 → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
26	25	alrimivv 1928	. . 3 ⊢ (𝜑 → ∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
27	26	alrimiv 1927	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
28		dffun2 6492	. . 3 ⊢ (Fun (𝐶 Func 𝐷) ↔ (Rel (𝐶 Func 𝐷) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ)))
29	28	biimpri 228	. 2 ⊢ ((Rel (𝐶 Func 𝐷) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ)) → Fun (𝐶 Func 𝐷))
30	1, 27, 29	sylancr 587	1 ⊢ (𝜑 → Fun (𝐶 Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∅c0 4284   class class class wbr 5092   × cxp 5617  Rel wrel 5624  Fun wfun 6476  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  Basecbs 17120  Hom chom 17172  Catccat 17570   Func cfunc 17761  ThinCatcthinc 49412

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755  df-ixp 8825  df-cat 17574  df-cid 17575  df-func 17765  df-thinc 49413

This theorem is referenced by: (None)