Theorem functhinc 49938

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 49502), and can be obtained from funcf2 17826, f002 49344, and ralrimivva 3182. (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 2739	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2739	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2739	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2739	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		functhinc.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		functhinc.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
12	11	thinccd 49913	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 12	isfunc 17822	. . . . 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 1100	. . . . 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 288	. . . 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 713	. . 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 49571	. . . . 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 776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
20		simprr 778	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
23	19, 20, 22	functhinclem2 49935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (((𝐹‘𝑣)𝐽(𝐹‘𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
24	2, 3, 4, 5, 11, 1, 18, 23	functhinclem1 49934	. . . . 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 284	. . . 4 ⊢ (𝜑 → (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ 𝐺 = 𝐾))
26	25	anbi1d 637	. . 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 280	. 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 49937	. 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 49287	1 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431  ∅c0 4261  ⟨cop 4561   class class class wbr 5072   × cxp 5616   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   ↑_m cmap 8763  Xcixp 8835  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622   Func cfunc 17812  ThinCatcthinc 49907

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ixp 8836  df-cat 17625  df-cid 17626  df-func 17816  df-thinc 49908

This theorem is referenced by:  functhincfun  49939  thincciso  49943  functermc  49998