Theorem idfullsubc 50268

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 18115. (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 18083	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
4	3	sseli 3927	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
5	2, 4	imaidfu2lem 50216	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
6	1, 5	eqtr4id 2815	. . 3 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → 𝐵 = ((1st ‘𝐼) “ (Base‘𝐷)))
7		eqid 2761	. . . . 5 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
8		eqid 2761	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2761	. . . . 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 18085	. . . . . 6 ⊢ Rel (𝐷 Full 𝐸)
11		1st2ndbr 8053	. . . . . 6 ⊢ ((Rel (𝐷 Full 𝐸) ∧ 𝐼 ∈ (𝐷 Full 𝐸)) → (1st ‘𝐼)(𝐷 Full 𝐸)(2nd ‘𝐼))
12	10, 11	mpan 703	. . . . 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 50258	. . . 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 3965	. 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 5683	. . . 4 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 × 𝐵) = (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))))
20	19	reseq2d 5970	. . 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 50218	. . 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 2806	. 2 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻)
24	17, 23	jca 521	1 ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  {csn 4584  ∪ ciun 4951   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654  Rel wrel 5656   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Hom chom 17439  Hom_f chomf 17840   Func cfunc 18029  id_funccidfu 18030   Full cful 18079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-ixp 8926  df-cat 17842  df-cid 17843  df-homf 17844  df-func 18033  df-idfu 18034  df-full 18081

This theorem is used by: (None)