Theorem idsubc 49047

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 2735	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		idsubc.h	. . 3 ⊢ 𝐻 = (Homf ‘𝐷)
5		eqid 2735	. . 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 49016	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu2 49018	. 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 2735	. . 3 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
9	2	func1st2nd 48991	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
10		f1oi 6855	. . . . . 6 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
11		dff1o3 6823	. . . . . 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 2736	. . . . . . 7 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Base‘𝐷) = (Base‘𝐷))
15	1, 2, 14	idfu1sta 49008	. . . . . 6 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
16	15	cnveqd 5855	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
17	16	funeqd 6557	. . . 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 49044	. 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 2834	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {csn 4601  ∪ ciun 4967   I cid 5547   × cxp 5652  ^◡ccnv 5653   ↾ cres 5656   “ cima 5657  Fun wfun 6524  –onto→wfo 6528  –1-1-onto→wf1o 6529  ‘cfv 6530  (class class class)co 7403   ∈ cmpo 7405  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17226  Hom chom 17280  Hom_f chomf 17676  Subcatcsubc 17820   Func cfunc 17865  id_funccidfu 17866

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-map 8840  df-pm 8841  df-ixp 8910  df-cat 17678  df-cid 17679  df-homf 17680  df-ssc 17821  df-subc 17823  df-func 17869  df-idfu 17870

This theorem is referenced by: (None)