Theorem functhincfun 50557

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 18037	. 2 ⊢ Rel (𝐶 Func 𝐷)
2		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓(𝐶 Func 𝐷)𝑔)
3		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
5		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		functhincfun.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝐶 ∈ Cat)
9		functhincfun.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
10	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝐷 ∈ ThinCat)
11	3, 4, 2	funcf1 18041	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
12		eqid 2760	. . . . . . . 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 789	. . . . . . . . . . 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 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
15		simprr 785	. . . . . . . . . . 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 18043	. . . . . . . . . 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 49964	. . . . . . . . 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 3205	. . . . . . . 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 50556	. . . . . . 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 235	. . . . . 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 785	. . . . . . 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 50556	. . . . . . 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 235	. . . . . 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 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑔 = ℎ)
25	24	ex 418	. . . 4 ⊢ (𝜑 → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
26	25	alrimivv 1960	. . 3 ⊢ (𝜑 → ∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
27	26	alrimiv 1959	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
28		dffun2 6548	. . 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 231	. 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 599	1 ⊢ (𝜑 → Fun (𝐶 Func 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∅c0 4278   class class class wbr 5102   × cxp 5648  Rel wrel 5655  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17387  Hom chom 17439  Catccat 17838   Func cfunc 18029  ThinCatcthinc 50525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-ixp 8926  df-cat 17842  df-cid 17843  df-func 18033  df-thinc 50526

This theorem is used by: (None)