Theorem idsubc 49260

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 2731	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		idsubc.h	. . 3 ⊢ 𝐻 = (Homf ‘𝐷)
5		eqid 2731	. . 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 49209	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu2 49211	. 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 2731	. . 3 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
9	2	func1st2nd 49176	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
10		f1oi 6801	. . . . . 6 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
11		dff1o3 6769	. . . . . 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 2732	. . . . . . 7 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Base‘𝐷) = (Base‘𝐷))
15	1, 2, 14	idfu1sta 49201	. . . . . 6 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
16	15	cnveqd 5814	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
17	16	funeqd 6503	. . . 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 49256	. 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 2831	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {csn 4573  ∪ ciun 4939   I cid 5508   × cxp 5612  ^◡ccnv 5613   ↾ cres 5616   “ cima 5617  Fun wfun 6475  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17120  Hom chom 17172  Hom_f chomf 17572  Subcatcsubc 17716   Func cfunc 17761  id_funccidfu 17762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-pm 8753  df-ixp 8822  df-cat 17574  df-cid 17575  df-homf 17576  df-ssc 17717  df-subc 17719  df-func 17765  df-idfu 17766

This theorem is referenced by: (None)