Theorem imasubc3 49515

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 2737	. . 3 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
6	1, 2, 3, 4, 5	imassc 49512	. 2 ⊢ (𝜑 → 𝐾 ⊆cat (Homf ‘𝐸))
7	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐹(𝐷 Func 𝐸)𝐺)
8		eqid 2737	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
10	1, 2, 3, 7, 8, 9	imaid 49513	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((Id‘𝐸)‘𝑎) ∈ (𝑎𝐾𝑎))
11	4	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹(𝐷 Func 𝐸)𝐺)
12		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
13		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
14		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
15	12, 13, 4	funcf1 17802	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
16		imasubc3.f	. . . . . . . . 9 ⊢ (𝜑 → Fun ◡𝐹)
17		df-f1 6505	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝐷)–1-1→(Base‘𝐸) ↔ (𝐹:(Base‘𝐷)⟶(Base‘𝐸) ∧ Fun ◡𝐹))
18	15, 16, 17	sylanbrc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
19	18	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝐹:(Base‘𝐷)–1-1→(Base‘𝐸))
20		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑎 ∈ 𝑆)
21		simplrl 777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑏 ∈ 𝑆)
22		simplrr 778	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑐 ∈ 𝑆)
23		simprl 771	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → 𝑓 ∈ (𝑎𝐾𝑏))
24		simprr 773	. . . . . . 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 49514	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑎𝐾𝑏) ∧ 𝑔 ∈ (𝑏𝐾𝑐))) → (𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
26	25	ralrimivva 3181	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆)) → ∀𝑓 ∈ (𝑎𝐾𝑏)∀𝑔 ∈ (𝑏𝐾𝑐)(𝑔(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)𝑓) ∈ (𝑎𝐾𝑐))
27	26	ralrimivva 3181	. . . 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 3130	. 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 49439	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
31		relfunc 17798	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
32	31	brrelex1i 5688	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
33	4, 32	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
34	33, 33, 3	imasubclem2 49464	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
35	5, 8, 14, 30, 34	issubc2 17772	. 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 714	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  {csn 4582  ⟨cop 4588  ∪ ciun 4948   class class class wbr 5100   × cxp 5630  ^◡ccnv 5631   “ cima 5635  Fun wfun 6494  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  Hom chom 17200  compcco 17201  Idccid 17600  Hom_f chomf 17601   ⊆_cat cssc 17743  Subcatcsubc 17745   Func cfunc 17790

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-pm 8778  df-ixp 8848  df-cat 17603  df-cid 17604  df-homf 17605  df-ssc 17746  df-subc 17748  df-func 17794

This theorem is referenced by:  idsubc  49519