Theorem idsubc 49153

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 2730	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		idsubc.h	. . 3 ⊢ 𝐻 = (Homf ‘𝐷)
5		eqid 2730	. . 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 49102	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu2 49104	. 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 2730	. . 3 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
9	2	func1st2nd 49069	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
10		f1oi 6841	. . . . . 6 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
11		dff1o3 6809	. . . . . 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 2731	. . . . . . 7 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (Base‘𝐷) = (Base‘𝐷))
15	1, 2, 14	idfu1sta 49094	. . . . . 6 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
16	15	cnveqd 5842	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
17	16	funeqd 6541	. . . 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 49149	. 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 2829	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4592  ∪ ciun 4958   I cid 5535   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   “ cima 5644  Fun wfun 6508  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17186  Hom chom 17238  Hom_f chomf 17634  Subcatcsubc 17778   Func cfunc 17823  id_funccidfu 17824

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-pm 8805  df-ixp 8874  df-cat 17636  df-cid 17637  df-homf 17638  df-ssc 17779  df-subc 17781  df-func 17827  df-idfu 17828

This theorem is referenced by: (None)