Theorem functhincfun 49837

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 17800	. 2 ⊢ Rel (𝐶 Func 𝐷)
2		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓(𝐶 Func 𝐷)𝑔)
3		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
5		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2737	. . . . . . . 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 17804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
12		eqid 2737	. . . . . . . 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 777	. . . . . . . . . . 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 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
15		simprr 773	. . . . . . . . . . 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 17806	. . . . . . . . . 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 49242	. . . . . . . . 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 3181	. . . . . . . 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 49836	. . . . . . 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 773	. . . . . . 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 49836	. . . . . . 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 2775	. . . . 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 1930	. . 3 ⊢ (𝜑 → ∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
27	26	alrimiv 1929	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝐶 Func 𝐷)𝑔 ∧ 𝑓(𝐶 Func 𝐷)ℎ) → 𝑔 = ℎ))
28		dffun2 6512	. . 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 588	1 ⊢ (𝜑 → Fun (𝐶 Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∅c0 4287   class class class wbr 5100   × cxp 5632  Rel wrel 5639  Fun wfun 6496  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  Basecbs 17150  Hom chom 17202  Catccat 17601   Func cfunc 17792  ThinCatcthinc 49805

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 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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-rmo 3352  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 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-map 8779  df-ixp 8850  df-cat 17605  df-cid 17606  df-func 17796  df-thinc 49806

This theorem is referenced by: (None)