Theorem imasubc2 49339

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 2734	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
6		eqid 2734	. . . 4 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
7	1, 2, 3, 4, 5, 6	imasubc 49338	. . 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 17830	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
10	9	ssbri 5141	. . . . 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 49267	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	7	simp2d 1143	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
14	5, 6, 12, 13	fullsubc 17772	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐸))
15	8, 14	eqeltrrd 2835	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  {csn 4578  ∪ ciun 4944   class class class wbr 5096   × cxp 5620  ^◡ccnv 5621   ↾ cres 5624   “ cima 5625   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17134  Hom chom 17186  Hom_f chomf 17587  Subcatcsubc 17731   Func cfunc 17776   Full cful 17826

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-pm 8764  df-ixp 8834  df-cat 17589  df-cid 17590  df-homf 17591  df-ssc 17732  df-subc 17734  df-func 17780  df-full 17828

This theorem is referenced by: (None)