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Theorem functhinc 46287
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 46254). (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 2740 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
7 eqid 2740 . . . . . 6 (Id‘𝐸) = (Id‘𝐸)
8 eqid 2740 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
9 eqid 2740 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
10 functhinc.d . . . . . 6 (𝜑𝐷 ∈ Cat)
11 functhinc.e . . . . . . 7 (𝜑𝐸 ∈ ThinCat)
1211thinccd 46267 . . . . . 6 (𝜑𝐸 ∈ Cat)
132, 3, 4, 5, 6, 7, 8, 9, 10, 12isfunc 17569 . . . . 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 1094 . . . . 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 704 . . 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 46260 . . . . 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 768 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑣𝐵)
20 simprr 770 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑢𝐵)
21 functhinc.1 . . . . . . . 8 (𝜑 → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2221adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2319, 20, 22functhinclem2 46284 . . . . . 6 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (((𝐹𝑣)𝐽(𝐹𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
242, 3, 4, 5, 11, 1, 18, 23functhinclem1 46283 . . . . 5 (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣𝐵𝑢𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹𝑣)𝐽(𝐹𝑢))) ↔ 𝐺 = 𝐾))
2517, 24syl5bb 283 . . . 4 (𝜑 → (𝐺X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑐))𝐽(𝐹‘(2nd𝑐))) ↑m (𝐻𝑐)) ↔ 𝐺 = 𝐾))
2625anbi1d 630 . . 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 278 . 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 46286 . 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 46103 1 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  wral 3066  Vcvv 3431  c0 4262  cop 4573   class class class wbr 5079   × cxp 5587   Fn wfn 6426  wf 6427  cfv 6431  (class class class)co 7269  cmpo 7271  1st c1st 7816  2nd c2nd 7817  m cmap 8590  Xcixp 8660  Basecbs 16902  Hom chom 16963  compcco 16964  Catccat 17363  Idccid 17364   Func cfunc 17559  ThinCatcthinc 46261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7818  df-2nd 7819  df-map 8592  df-ixp 8661  df-cat 17367  df-cid 17368  df-func 17563  df-thinc 46262
This theorem is referenced by:  thincciso  46291
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