Theorem imasubc3 49343

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 2734	. . 3 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
6	1, 2, 3, 4, 5	imassc 49340	. 2 ⊢ (𝜑 → 𝐾 ⊆cat (Homf ‘𝐸))
7	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹(𝐷 Func 𝐸)𝐺)
8		eqid 2734	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
10	1, 2, 3, 7, 8, 9	imaid 49341	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎))
11	4	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹(𝐷 Func 𝐸)𝐺)
12		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
13		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
14		eqid 2734	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
15	12, 13, 4	funcf1 17788	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
16		imasubc3.f	. . . . . . . . 9 ⊢ (𝜑 → Fun ◡𝐹)
17		df-f1 6495	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝐷)–1-1→(Base‘𝐸) ↔ (𝐹:(Base‘𝐷)⟶(Base‘𝐸) ∧ Fun ◡𝐹))
18	15, 16, 17	sylanbrc 583	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
19	18	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
20		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑎 ∈ 𝑆)
21		simplrl 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑏 ∈ 𝑆)
22		simplrr 777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑐 ∈ 𝑆)
23		simprl 770	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑓 ∈ (𝑎𝐾𝑏))
24		simprr 772	. . . . . . 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 49342	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
26	25	ralrimivva 3177	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) → ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
27	26	ralrimivva 3177	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑆 ∀𝑐 ∈ 𝑆 ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
28	10, 27	jca 511	. . 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 3126	. 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 49267	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
31		relfunc 17784	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
32	31	brrelex1i 5678	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
33	4, 32	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
34	33, 33, 3	imasubclem2 49292	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
35	5, 8, 14, 30, 34	issubc2 17758	. 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 713	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  {csn 4578  ⟨cop 4584  ∪ ciun 4944   class class class wbr 5096   × cxp 5620  ^◡ccnv 5621   “ cima 5625  Fun wfun 6484  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17134  Hom chom 17186  compcco 17187  Idccid 17586  Hom_f chomf 17587   ⊆_cat cssc 17729  Subcatcsubc 17731   Func cfunc 17776

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

This theorem is referenced by:  idsubc  49347