Theorem functhinc 49097

Description: A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 48901), and can be obtained from funcf2 17913, f002 48763, and ralrimivva 3202. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
functhinc.b	⊢ 𝐵 = (Base‘𝐷)
functhinc.c	⊢ 𝐶 = (Base‘𝐸)
functhinc.h	⊢ 𝐻 = (Hom ‘𝐷)
functhinc.j	⊢ 𝐽 = (Hom ‘𝐸)
functhinc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
functhinc.e	⊢ (𝜑 → 𝐸 ∈ ThinCat)
functhinc.f	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
functhinc.k	⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
functhinc.1	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))

Assertion

Ref	Expression
functhinc	⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))

Distinct variable groups:   𝑤,𝐹,𝑧   𝑥,𝐹,𝑦   𝑤,𝐻,𝑧   𝑥,𝐻,𝑦   𝑤,𝐽,𝑧   𝑥,𝐽,𝑦   𝑤,𝐵,𝑧   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝐺(𝑥,𝑦,𝑧,𝑤)   𝐾(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem functhinc

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		functhinc.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
2		functhinc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
3		functhinc.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
4		functhinc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐷)
5		functhinc.j	. . . . . 6 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2737	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2737	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2737	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2737	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		functhinc.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		functhinc.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
12	11	thinccd 49073	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 12	isfunc 17909	. . . . 5 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))
14		3anass 1095	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))
15	13, 14	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))))
16	1, 15	mpbirand 707	. . 3 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))
17		funcf2lem 48914	. . . . 5 ⊢ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹‘𝑣)𝐽(𝐹‘𝑢))))
18		functhinc.k	. . . . . 6 ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
19		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
20		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
21		functhinc.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
23	19, 20, 22	functhinclem2 49094	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (((𝐹‘𝑣)𝐽(𝐹‘𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
24	2, 3, 4, 5, 11, 1, 18, 23	functhinclem1 49093	. . . . 5 ⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹‘𝑣)𝐽(𝐹‘𝑢))) ↔ 𝐺 = 𝐾))
25	17, 24	bitrid 283	. . . 4 ⊢ (𝜑 → (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ 𝐺 = 𝐾))
26	25	anbi1d 631	. . 3 ⊢ (𝜑 → ((𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐺 = 𝐾 ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))
27	16, 26	bitrd 279	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺 = 𝐾 ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))
28	2, 3, 4, 5, 10, 11, 1, 18, 21, 6, 7, 8, 9	functhinclem4 49096	. 2 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))
29	27, 28	mpbiran3d 48717	1 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ∅c0 4333  ⟨cop 4632   class class class wbr 5143   × cxp 5683   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013   ↑_m cmap 8866  Xcixp 8937  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708   Func cfunc 17899  ThinCatcthinc 49067

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-func 17903  df-thinc 49068

This theorem is referenced by:  functhincfun  49098  thincciso  49102  functermc  49140