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Theorem functhinc 47665
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 47632). (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 2733 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
7 eqid 2733 . . . . . 6 (Id‘𝐸) = (Id‘𝐸)
8 eqid 2733 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
9 eqid 2733 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
10 functhinc.d . . . . . 6 (𝜑𝐷 ∈ Cat)
11 functhinc.e . . . . . . 7 (𝜑𝐸 ∈ ThinCat)
1211thinccd 47645 . . . . . 6 (𝜑𝐸 ∈ Cat)
132, 3, 4, 5, 6, 7, 8, 9, 10, 12isfunc 17814 . . . . 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 1096 . . . . 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 706 . . 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 47638 . . . . 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 770 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑣𝐵)
20 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑢𝐵)
21 functhinc.1 . . . . . . . 8 (𝜑 → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2221adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → ∀𝑧𝐵𝑤𝐵 (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
2319, 20, 22functhinclem2 47662 . . . . . 6 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (((𝐹𝑣)𝐽(𝐹𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
242, 3, 4, 5, 11, 1, 18, 23functhinclem1 47661 . . . . 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 47664 . 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 47482 1 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  c0 4323  cop 4635   class class class wbr 5149   × cxp 5675   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974  m cmap 8820  Xcixp 8891  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609   Func cfunc 17804  ThinCatcthinc 47639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-ixp 8892  df-cat 17612  df-cid 17613  df-func 17808  df-thinc 47640
This theorem is referenced by:  thincciso  47669
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