Theorem imassc 49135

Description: An image of a functor satisfies the subcategory subset relation. (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)
imassc.j	⊢ 𝐽 = (Homf ‘𝐸)

Assertion

Ref	Expression
imassc	⊢ (𝜑 → 𝐾 ⊆cat 𝐽)

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)   𝐽(𝑥,𝑦,𝑝)   𝐾(𝑥,𝑦,𝑝)

Proof of Theorem imassc

Dummy variables 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasubc.s	. . 3 ⊢ 𝑆 = (𝐹 “ 𝐴)
2		eqid 2729	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2729	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		imassc.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	2, 3, 4	funcf1 17808	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
6	5	fimassd 6691	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (Base‘𝐸))
7	1, 6	eqsstrid 3982	. 2 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
8		imasubc.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐷)
9		eqid 2729	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
10	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝐹(𝐷 Func 𝐸)𝐺)
11	2, 3, 10	funcf1 17808	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
12	11	ffnd 6671	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝐹 Fn (Base‘𝐷))
13		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑚 ∈ (◡𝐹 “ {𝑧}))
14		fniniseg 7014	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑧}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧)))
15	14	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝐷) ∧ 𝑚 ∈ (◡𝐹 “ {𝑧})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧))
16	12, 13, 15	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑧))
17	16	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑚 ∈ (Base‘𝐷))
18		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑛 ∈ (◡𝐹 “ {𝑤}))
19		fniniseg 7014	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑛 ∈ (◡𝐹 “ {𝑤}) ↔ (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤)))
20	19	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝐷) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤})) → (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤))
21	12, 18, 20	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑛 ∈ (Base‘𝐷) ∧ (𝐹‘𝑛) = 𝑤))
22	21	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → 𝑛 ∈ (Base‘𝐷))
23	2, 8, 9, 10, 17, 22	funcf2 17810	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)))
24	23	fimassd 6691	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ ((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)))
25	16	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝐹‘𝑚) = 𝑧)
26	21	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → (𝐹‘𝑛) = 𝑤)
27	25, 26	oveq12d 7387	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → ((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑛)) = (𝑧(Hom ‘𝐸)𝑤))
28	24, 27	sseqtrd 3980	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑚 ∈ (◡𝐹 “ {𝑧}) ∧ 𝑛 ∈ (◡𝐹 “ {𝑤}))) → ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
29	28	ralrimivva 3178	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ∀𝑚 ∈ (◡𝐹 “ {𝑧})∀𝑛 ∈ (◡𝐹 “ {𝑤})((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
30		iunss 5004	. . . . . 6 ⊢ (∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤) ↔ ∀𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
31		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑛⟩))
32		df-ov 7372	. . . . . . . . . 10 ⊢ (𝑚𝐺𝑛) = (𝐺‘⟨𝑚, 𝑛⟩)
33	31, 32	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑛))
34		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑛⟩))
35		df-ov 7372	. . . . . . . . . 10 ⊢ (𝑚𝐻𝑛) = (𝐻‘⟨𝑚, 𝑛⟩)
36	34, 35	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑛))
37	33, 36	imaeq12d 6021	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)))
38	37	sseq1d 3975	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑛⟩ → (((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤) ↔ ((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ (𝑧(Hom ‘𝐸)𝑤)))
39	38	ralxp 5795	. . . . . 6 ⊢ (∀𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤) ↔ ∀𝑚 ∈ (◡𝐹 “ {𝑧})∀𝑛 ∈ (◡𝐹 “ {𝑤})((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
40	30, 39	bitri 275	. . . . 5 ⊢ (∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤) ↔ ∀𝑚 ∈ (◡𝐹 “ {𝑧})∀𝑛 ∈ (◡𝐹 “ {𝑤})((𝑚𝐺𝑛) “ (𝑚𝐻𝑛)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
41	29, 40	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)) ⊆ (𝑧(Hom ‘𝐸)𝑤))
42		relfunc 17804	. . . . . . . 8 ⊢ Rel (𝐷 Func 𝐸)
43	42	brrelex1i 5687	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
44	4, 43	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
45	44	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐹 ∈ V)
46		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
47		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ 𝑆)
48		imasubc.k	. . . . 5 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
49	45, 45, 46, 47, 48	imasubclem3 49088	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐾𝑤) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑧}) × (◡𝐹 “ {𝑤}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
50		imassc.j	. . . . 5 ⊢ 𝐽 = (Homf ‘𝐸)
51	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐸))
52	51, 46	sseldd 3944	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐸))
53	51, 47	sseldd 3944	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ (Base‘𝐸))
54	50, 3, 9, 52, 53	homfval 17633	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧(Hom ‘𝐸)𝑤))
55	41, 49, 54	3sstr4d 3999	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐾𝑤) ⊆ (𝑧𝐽𝑤))
56	55	ralrimivva 3178	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐾𝑤) ⊆ (𝑧𝐽𝑤))
57	44, 44, 48	imasubclem2 49087	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
58	50, 3	homffn 17634	. . . 4 ⊢ 𝐽 Fn ((Base‘𝐸) × (Base‘𝐸))
59	58	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 Fn ((Base‘𝐸) × (Base‘𝐸)))
60		fvexd 6855	. . 3 ⊢ (𝜑 → (Base‘𝐸) ∈ V)
61	57, 59, 60	isssc 17762	. 2 ⊢ (𝜑 → (𝐾 ⊆cat 𝐽 ↔ (𝑆 ⊆ (Base‘𝐸) ∧ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐾𝑤) ⊆ (𝑧𝐽𝑤))))
62	7, 56, 61	mpbir2and 713	1 ⊢ (𝜑 → 𝐾 ⊆cat 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  {csn 4585  ⟨cop 4591  ∪ ciun 4951   class class class wbr 5102   × cxp 5629  ^◡ccnv 5630   “ cima 5634   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Basecbs 17155  Hom chom 17207  Hom_f chomf 17607   ⊆_cat cssc 17749   Func cfunc 17796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-ixp 8848  df-homf 17611  df-ssc 17752  df-func 17800

This theorem is referenced by:  imasubc3  49138