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Theorem functhinc 47743
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 47710). (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 2732 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
7 eqid 2732 . . . . . 6 (Id‘𝐸) = (Id‘𝐸)
8 eqid 2732 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
9 eqid 2732 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
10 functhinc.d . . . . . 6 (𝜑𝐷 ∈ Cat)
11 functhinc.e . . . . . . 7 (𝜑𝐸 ∈ ThinCat)
1211thinccd 47723 . . . . . 6 (𝜑𝐸 ∈ Cat)
132, 3, 4, 5, 6, 7, 8, 9, 10, 12isfunc 17816 . . . . 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 286 . . . 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 705 . . 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 47716 . . . . 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 769 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → 𝑣𝐵)
20 simprr 771 . . . . . . 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 47740 . . . . . 6 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (((𝐹𝑣)𝐽(𝐹𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
242, 3, 4, 5, 11, 1, 18, 23functhinclem1 47739 . . . . 5 (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣𝐵𝑢𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹𝑣)𝐽(𝐹𝑢))) ↔ 𝐺 = 𝐾))
2517, 24bitrid 282 . . . 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 47742 . 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 47560 1 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  c0 4322  cop 4634   class class class wbr 5148   × cxp 5674   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976  m cmap 8822  Xcixp 8893  Basecbs 17146  Hom chom 17210  compcco 17211  Catccat 17610  Idccid 17611   Func cfunc 17806  ThinCatcthinc 47717
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-ixp 8894  df-cat 17614  df-cid 17615  df-func 17810  df-thinc 47718
This theorem is referenced by:  thincciso  47747
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