Theorem idfullsubc 49549

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 17878. (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 17846	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
4	3	sseli 3931	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
5	2, 4	imaidfu2lem 49497	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
6	1, 5	eqtr4id 2791	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐵 = ((1st ‘𝐼) “ (Base‘𝐷)))
7		eqid 2737	. . . . 5 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
8		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2737	. . . . 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 17848	. . . . . 6 ⊢ Rel (𝐷 Full 𝐸)
11		1st2ndbr 7998	. . . . . 6 ⊢ ((Rel (𝐷 Full 𝐸) ∧ 𝐼 ∈ (𝐷 Full 𝐸)) → (1st ‘𝐼)(𝐷 Full 𝐸)(2nd ‘𝐼))
12	10, 11	mpan 691	. . . . 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 49539	. . . 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 1144	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) ⊆ 𝐶)
17	6, 16	eqsstrd 3970	. 2 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐵 ⊆ 𝐶)
18	15	simp3d 1145	. . 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 5666	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 × 𝐵) = (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))))
20	19	reseq2d 5948	. . 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 49499	. . 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 2782	. 2 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻)
24	17, 23	jca 511	1 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  {csn 4582  ∪ ciun 4948   class class class wbr 5100   × cxp 5632  ^◡ccnv 5633   ↾ cres 5636   “ cima 5637  Rel wrel 5639   Fn wfn 6497  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1^st c1st 7943  2^nd c2nd 7944  Basecbs 17150  Hom chom 17202  Hom_f chomf 17603   Func cfunc 17792  id_funccidfu 17793   Full cful 17842

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-map 8779  df-ixp 8850  df-cat 17605  df-cid 17606  df-homf 17607  df-func 17796  df-idfu 17797  df-full 17844

This theorem is referenced by: (None)