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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 (𝜑𝐷 ∈ Cat)
functhinc.e (𝜑𝐸 ∈ ThinCat)
functhinc.f (𝜑𝐹:𝐵𝐶)
functhinc.k 𝐾 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))))
functhinc.1 (𝜑 → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
Assertion
Ref Expression
functhinc (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐺 = 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.
StepHypRef Expression
1 functhinc.f . . . 4 (𝜑𝐹:𝐵𝐶)
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 (𝜑𝐷 ∈ Cat)
11 functhinc.e . . . . . . 7 (𝜑𝐸 ∈ ThinCat)
1211thinccd 49073 . . . . . 6 (𝜑𝐸 ∈ Cat)
132, 3, 4, 5, 6, 7, 8, 9, 10, 12isfunc 17909 . . . . 5 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))))
14 3anass 1095 . . . . 5 ((𝐹:𝐵𝐶𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐹:𝐵𝐶 ∧ (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))))
1513, 14bitrdi 287 . . . 4 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶 ∧ (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓)))))))
161, 15mpbirand 707 . . 3 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))))
17 funcf2lem 48914 . . . . 5 (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣𝐵𝑢𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹𝑣)𝐽(𝐹𝑢))))
18 functhinc.k . . . . . 6 𝐾 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))))
19 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑣𝐵)
20 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑢𝐵)
21 functhinc.1 . . . . . . . 8 (𝜑 → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2221adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2319, 20, 22functhinclem2 49094 . . . . . 6 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (((𝐹𝑣)𝐽(𝐹𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
242, 3, 4, 5, 11, 1, 18, 23functhinclem1 49093 . . . . 5 (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣𝐵𝑢𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹𝑣)𝐽(𝐹𝑢))) ↔ 𝐺 = 𝐾))
2517, 24bitrid 283 . . . 4 (𝜑 → (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ↔ 𝐺 = 𝐾))
2625anbi1d 631 . . 3 (𝜑 → ((𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐺 = 𝐾 ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))))
2716, 26bitrd 279 . 2 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺 = 𝐾 ∧ ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))))
282, 3, 4, 5, 10, 11, 1, 18, 21, 6, 7, 8, 9functhinclem4 49096 . 2 ((𝜑𝐺 = 𝐾) → ∀𝑎𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹𝑎)) ∧ ∀𝑏𝐵𝑐𝐵𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝐸)(𝐹𝑐))((𝑎𝐺𝑏)‘𝑓))))
2927, 28mpbiran3d 48717 1 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐺 = 𝐾))
Colors of variables: wff setvar class
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  1st c1st 8012  2nd 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
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