Theorem idsubc 49635

Description: The source category of an inclusion functor is a subcategory of the target category. See also Remark 4.4 in [Adamek] p. 49. (Contributed by Zhi Wang, 10-Nov-2025.)

Hypotheses

Ref	Expression
idfth.i	⊢ 𝐼 = (idfunc‘𝐶)
idsubc.h	⊢ 𝐻 = (Homf ‘𝐷)

Assertion

Ref	Expression
idsubc	⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

Proof of Theorem idsubc

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

Step	Hyp	Ref	Expression
1		idfth.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐶)
2		id 22	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
3		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		idsubc.h	. . 3 ⊢ 𝐻 = (Homf ‘𝐷)
5		eqid 2736	. . 3 ⊢ (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)))
6	1, 2	imaidfu2lem 49584	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu2 49586	. 2 ⊢ (𝐼 ∈ (𝐷 Func 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))))
8		eqid 2736	. . 3 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
9	2	func1st2nd 49551	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
10		f1oi 6818	. . . . . 6 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
11		dff1o3 6786	. . . . . 6 ⊢ (( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷) ↔ (( I ↾ (Base‘𝐷)):(Base‘𝐷)–onto→(Base‘𝐷) ∧ Fun ◡( I ↾ (Base‘𝐷))))
12	10, 11	mpbi 230	. . . . 5 ⊢ (( I ↾ (Base‘𝐷)):(Base‘𝐷)–onto→(Base‘𝐷) ∧ Fun ◡( I ↾ (Base‘𝐷)))
13	12	simpri 485	. . . 4 ⊢ Fun ◡( I ↾ (Base‘𝐷))
14		eqidd 2737	. . . . . . 7 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Base‘𝐷) = (Base‘𝐷))
15	1, 2, 14	idfu1sta 49576	. . . . . 6 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
16	15	cnveqd 5830	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
17	16	funeqd 6520	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Fun ◡(1st ‘𝐼) ↔ Fun ◡( I ↾ (Base‘𝐷))))
18	13, 17	mpbiri 258	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ◡(1st ‘𝐼))
19	8, 3, 5, 9, 18	imasubc3 49631	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ ((Hom ‘𝐷)‘𝑝))) ∈ (Subcat‘𝐸))
20	7, 19	eqeltrd 2836	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4567  ∪ ciun 4933   I cid 5525   × cxp 5629  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634  Fun wfun 6492  –onto→wfo 6496  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231  Hom_f chomf 17632  Subcatcsubc 17776   Func cfunc 17821  id_funccidfu 17822

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-pm 8776  df-ixp 8846  df-cat 17634  df-cid 17635  df-homf 17636  df-ssc 17777  df-subc 17779  df-func 17825  df-idfu 17826

This theorem is referenced by: (None)