Theorem functhincfun 49560

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 2731	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
5		eqid 2731	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2731	. . . . . . . 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 2731	. . . . . . . 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 48964	. . . . . . . . 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 3175	. . . . . . . 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 49559	. . . . . . 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 49559	. . . . . . 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 2769	. . . . 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 1929	. . 3 ⊢ (𝜑 → ∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
27	26	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
28		dffun2 6491	. . 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 1539   = wceq 1541   ∈ wcel 2111  ∅c0 4280   class class class wbr 5089   × cxp 5612  Rel wrel 5619  Fun wfun 6475  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17120  Hom chom 17172  Catccat 17570   Func cfunc 17761  ThinCatcthinc 49528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-ixp 8822  df-cat 17574  df-cid 17575  df-func 17765  df-thinc 49529

This theorem is referenced by: (None)