Theorem idsubc 50089

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 23	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
3		eqid 2760	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		idsubc.h	. . 3 ⊢ 𝐻 = (Homf ‘𝐷)
5		eqid 2760	. . 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 50038	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu2 50040	. 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 2760	. . 3 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
9	2	func1st2nd 50005	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
10		f1oi 6857	. . . . . 6 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
11		dff1o3 6825	. . . . . 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 233	. . . . 5 ⊢ (( I ↾ (Base‘𝐷)):(Base‘𝐷)–onto→(Base‘𝐷) ∧ Fun ◡( I ↾ (Base‘𝐷)))
13	12	simpri 491	. . . 4 ⊢ Fun ◡( I ↾ (Base‘𝐷))
14		eqidd 2761	. . . . . . 7 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Base‘𝐷) = (Base‘𝐷))
15	1, 2, 14	idfu1sta 50030	. . . . . 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 6555	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Fun ◡(1st ‘𝐼) ↔ Fun ◡( I ↾ (Base‘𝐷))))
18	13, 17	mpbiri 261	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ◡(1st ‘𝐼))
19	8, 3, 5, 9, 18	imasubc3 50085	. 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 2860	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4584  ∪ ciun 4951   I cid 5549   × cxp 5653  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  Fun wfun 6527  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17304  Hom chom 17356  Hom_f chomf 17757  Subcatcsubc 17901   Func cfunc 17946  id_funccidfu 17947

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8831  df-pm 8832  df-ixp 8908  df-cat 17759  df-cid 17760  df-homf 17761  df-ssc 17902  df-subc 17904  df-func 17950  df-idfu 17951

This theorem is used by: (None)