Theorem functhinc 49441

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 49005), and can be obtained from funcf2 17837, f002 48846, and ralrimivva 3181. (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 2730	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2730	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2730	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2730	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		functhinc.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		functhinc.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
12	11	thinccd 49416	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 12	isfunc 17833	. . . . 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 1094	. . . . 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 287	. . . 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 707	. . 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 49074	. . . . 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 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
20		simprr 772	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
23	19, 20, 22	functhinclem2 49438	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (((𝐹‘𝑣)𝐽(𝐹‘𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅))
24	2, 3, 4, 5, 11, 1, 18, 23	functhinclem1 49437	. . . . 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 283	. . . 4 ⊢ (𝜑 → (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ 𝐺 = 𝐾))
26	25	anbi1d 631	. . 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 279	. 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 49440	. 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 48789	1 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450  ∅c0 4299  ⟨cop 4598   class class class wbr 5110   × cxp 5639   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802  Xcixp 8873  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633   Func cfunc 17823  ThinCatcthinc 49410

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-ixp 8874  df-cat 17636  df-cid 17637  df-func 17827  df-thinc 49411

This theorem is referenced by:  functhincfun  49442  thincciso  49446  functermc  49501