Theorem imasubc2 49627

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 2736	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
6		eqid 2736	. . . 4 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
7	1, 2, 3, 4, 5, 6	imasubc 49626	. . 3 ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ (Base‘𝐸) ∧ ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾))
8	7	simp3d 1145	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾)
9		fullfunc 17875	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
10	9	ssbri 5130	. . . . 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 49555	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	7	simp2d 1144	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
14	5, 6, 12, 13	fullsubc 17817	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐸))
15	8, 14	eqeltrrd 2837	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  {csn 4567  ∪ ciun 4933   class class class wbr 5085   × cxp 5629  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Hom chom 17231  Hom_f chomf 17632  Subcatcsubc 17776   Func cfunc 17821   Full cful 17871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-pm 8776  df-ixp 8846  df-cat 17634  df-cid 17635  df-homf 17636  df-ssc 17777  df-subc 17779  df-func 17825  df-full 17873

This theorem is referenced by: (None)