Theorem idfullsubc 49967

Description: The source category of an inclusion functor is a full subcategory of the target category if the inclusion functor is full. Remark 4.4(2) in [Adamek] p. 49. See also ressffth 18001. (Contributed by Zhi Wang, 11-Nov-2025.)

Hypotheses

Ref	Expression
idfth.i	⊢ 𝐼 = (idfunc‘𝐶)
idsubc.h	⊢ 𝐻 = (Homf ‘𝐷)
idfullsubc.j	⊢ 𝐽 = (Homf ‘𝐸)
idfullsubc.b	⊢ 𝐵 = (Base‘𝐷)
idfullsubc.c	⊢ 𝐶 = (Base‘𝐸)

Assertion

Ref	Expression
idfullsubc	⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))

Proof of Theorem idfullsubc

Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfullsubc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		idfth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐶)
3		fullfunc 17969	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
4	3	sseli 3933	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
5	2, 4	imaidfu2lem 49915	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
6	1, 5	eqtr4id 2817	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐵 = ((1st ‘𝐼) “ (Base‘𝐷)))
7		eqid 2763	. . . . 5 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
8		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝))) = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝)))
10		relfull 17971	. . . . . 6 ⊢ Rel (𝐷 Full 𝐸)
11		1st2ndbr 8035	. . . . . 6 ⊢ ((Rel (𝐷 Full 𝐸) ∧ 𝐼 ∈ (𝐷 Full 𝐸)) → (1st ‘𝐼)(𝐷 Full 𝐸)(2nd ‘𝐼))
12	10, 11	mpan 702	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (1st ‘𝐼)(𝐷 Full 𝐸)(2nd ‘𝐼))
13		idfullsubc.c	. . . . 5 ⊢ 𝐶 = (Base‘𝐸)
14		idfullsubc.j	. . . . 5 ⊢ 𝐽 = (Homf ‘𝐸)
15	7, 8, 9, 12, 13, 14	imasubc 49957	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝))) Fn (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))) ∧ ((1st ‘𝐼) “ (Base‘𝐷)) ⊆ 𝐶 ∧ (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝)))))
16	15	simp2d 1161	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) ⊆ 𝐶)
17	6, 16	eqsstrd 3971	. 2 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐵 ⊆ 𝐶)
18	15	simp3d 1162	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝))))
19	6	sqxpeqd 5693	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 × 𝐵) = (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))))
20	19	reseq2d 5978	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐽 ↾ (𝐵 × 𝐵)) = (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))))
21		idsubc.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐷)
22	2, 4, 8, 21, 9, 5	imaidfu2 49917	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐻 = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝))))
23	18, 20, 22	3eqtr4d 2808	. 2 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻)
24	17, 23	jca 520	1 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ∪ ciun 4956   class class class wbr 5109   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664  Rel wrel 5666   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17273  Hom chom 17325  Hom_f chomf 17726   Func cfunc 17915  id_funccidfu 17916   Full cful 17965

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-cat 17728  df-cid 17729  df-homf 17730  df-func 17919  df-idfu 17920  df-full 17967

This theorem is used by: (None)