Theorem imasubc2 49958

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 2762	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
6		eqid 2762	. . . 4 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
7	1, 2, 3, 4, 5, 6	imasubc 49957	. . 3 ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ (Base‘𝐸) ∧ ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾))
8	7	simp3d 1161	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾)
9		fullfunc 17971	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
10	9	ssbri 5155	. . . . 5 ⊢ (𝐹(𝐷 Full 𝐸)𝐺 → 𝐹(𝐷 Func 𝐸)𝐺)
11	4, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
12	11	funcrcl3 49886	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	7	simp2d 1160	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
14	5, 6, 12, 13	fullsubc 17913	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐸))
15	8, 14	eqeltrrd 2863	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ⊆ wss 3904  {csn 4588  ∪ ciun 4955   class class class wbr 5108   × cxp 5658  ^◡ccnv 5659   ↾ cres 5662   “ cima 5663   Fn wfn 6531  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17275  Hom chom 17327  Hom_f chomf 17728  Subcatcsubc 17872   Func cfunc 17917   Full cful 17967

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-map 8824  df-pm 8825  df-ixp 8894  df-cat 17730  df-cid 17731  df-homf 17732  df-ssc 17873  df-subc 17875  df-func 17921  df-full 17969

This theorem is used by: (None)