Theorem imasubc3 49954

Description: An image of a functor injective on objects is a 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)))
imassc.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
imasubc3.f	⊢ (𝜑 → Fun ◡𝐹)

Assertion

Ref	Expression
imasubc3	⊢ (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 imasubc3

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasubc.s	. . 3 ⊢ 𝑆 = (𝐹 “ 𝐴)
2		imasubc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
3		imasubc.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
4		imassc.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5		eqid 2763	. . 3 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
6	1, 2, 3, 4, 5	imassc 49951	. 2 ⊢ (𝜑 → 𝐾 ⊆cat (Homf ‘𝐸))
7	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹(𝐷 Func 𝐸)𝐺)
8		eqid 2763	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
9		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
10	1, 2, 3, 7, 8, 9	imaid 49952	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎))
11	4	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹(𝐷 Func 𝐸)𝐺)
12		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
13		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
14		eqid 2763	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
15	12, 13, 4	funcf1 17918	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
16		imasubc3.f	. . . . . . . . 9 ⊢ (𝜑 → Fun ◡𝐹)
17		df-f1 6541	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝐷)–1-1→(Base‘𝐸) ↔ (𝐹:(Base‘𝐷)⟶(Base‘𝐸) ∧ Fun ◡𝐹))
18	15, 16, 17	sylanbrc 594	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
19	18	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
20		simpllr 787	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑎 ∈ 𝑆)
21		simplrl 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑏 ∈ 𝑆)
22		simplrr 789	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑐 ∈ 𝑆)
23		simprl 782	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑓 ∈ (𝑎𝐾𝑏))
24		simprr 784	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑔 ∈ (𝑏𝐾𝑐))
25	1, 2, 3, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24	imaf1co 49953	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
26	25	ralrimivva 3208	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) → ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
27	26	ralrimivva 3208	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
28	10, 27	jca 520	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐)))
29	28	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐)))
30	4	funcrcl3 49878	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
31		relfunc 17914	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
32	31	brrelex1i 5717	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
33	4, 32	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
34	33, 33, 3	imasubclem2 49903	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
35	5, 8, 14, 30, 34	issubc2 17888	. 2 ⊢ (𝜑 → (𝐾 ∈ (Subcat‘𝐸) ↔ (𝐾 ⊆cat (Homf ‘𝐸) ∧ ∀𝑎 ∈ 𝑆 (((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎) ∧ ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐)))))
36	6, 29, 35	mpbir2and 725	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  {csn 4589  ⟨cop 4595  ∪ ciun 4956   class class class wbr 5109   × cxp 5659  ^◡ccnv 5660   “ cima 5664  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  Hom chom 17316  compcco 17317  Idccid 17716  Hom_f chomf 17717   ⊆_cat cssc 17859  Subcatcsubc 17861   Func cfunc 17906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-pm 8823  df-ixp 8892  df-cat 17719  df-cid 17720  df-homf 17721  df-ssc 17862  df-subc 17864  df-func 17910

This theorem is referenced by:  idsubc  49958