Theorem functhincfun 49198

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 17862	. 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 2734	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
5		eqid 2734	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2734	. . . . . . . 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 17866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
12		eqid 2734	. . . . . . . 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 17868	. . . . . . . . . 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 48726	. . . . . . . . 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 3185	. . . . . . . 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 49197	. . . . . . 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 49197	. . . . . . 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 2772	. . . . 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 1927	. . 3 ⊢ (𝜑 → ∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
27	26	alrimiv 1926	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
28		dffun2 6538	. . 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 1537   = wceq 1539   ∈ wcel 2107  ∅c0 4306   class class class wbr 5117   × cxp 5650  Rel wrel 5657  Fun wfun 6522  ‘cfv 6528  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17215  Hom chom 17269  Catccat 17663   Func cfunc 17854  ThinCatcthinc 49166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-map 8837  df-ixp 8907  df-cat 17667  df-cid 17668  df-func 17858  df-thinc 49167

This theorem is referenced by: (None)