Theorem imasubc2 49263

Description: An image of a full functor is a (full) subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imasubc.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imasubc.h	⊢ 𝐻 = (Hom ‘𝐷)
imasubc.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
imasubc.f	⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)

Assertion

Ref	Expression
imasubc2	⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝐸,𝑝   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥,𝑦,𝑝)   𝐷(𝑥,𝑦,𝑝)   𝑆(𝑝)   𝐸(𝑥,𝑦)   𝐾(𝑥,𝑦,𝑝)

Proof of Theorem imasubc2

Step	Hyp	Ref	Expression
1		imasubc.s	. . . 4 ⊢ 𝑆 = (𝐹 “ 𝐴)
2		imasubc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
3		imasubc.k	. . . 4 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
4		imasubc.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)
5		eqid 2731	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
6		eqid 2731	. . . 4 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
7	1, 2, 3, 4, 5, 6	imasubc 49262	. . 3 ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ (Base‘𝐸) ∧ ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾))
8	7	simp3d 1144	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾)
9		fullfunc 17815	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
10	9	ssbri 5134	. . . . 5 ⊢ (𝐹(𝐷 Full 𝐸)𝐺 → 𝐹(𝐷 Func 𝐸)𝐺)
11	4, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
12	11	funcrcl3 49191	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	7	simp2d 1143	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
14	5, 6, 12, 13	fullsubc 17757	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐸))
15	8, 14	eqeltrrd 2832	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  {csn 4573  ∪ ciun 4939   class class class wbr 5089   × cxp 5612  ^◡ccnv 5613   ↾ cres 5616   “ cima 5617   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17120  Hom chom 17172  Hom_f chomf 17572  Subcatcsubc 17716   Func cfunc 17761   Full cful 17811

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-full 17813

This theorem is referenced by: (None)