Theorem imaid 49991

Description: An image of a functor preserves the identity morphism. (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)
imaid.i	⊢ 𝐼 = (Id‘𝐸)
imaid.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
imaid	⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝐼,𝑝   𝑋,𝑝,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑝)   𝐴(𝑥, 𝑦, 𝑝)   𝐷(𝑥, 𝑦, 𝑝)   𝑆(𝑝)   𝐸(𝑥, 𝑦, 𝑝)   𝐼(𝑥, 𝑦)   𝐾(𝑥, 𝑦, 𝑝)

Proof of Theorem imaid

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
2		imasubc.s	. . . . . . 7 ⊢ 𝑆 = (𝐹 “ 𝐴)
3	1, 2	eleqtrdi 2875	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
4		inisegn0a 49673	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑋}) ≠ ∅)
5	3, 4	syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ ∅)
6		n0 4307	. . . . 5 ⊢ ((◡𝐹 “ {𝑋}) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
7	5, 6	sylib 221	. . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
8		fveq2 6885	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑚⟩))
9		df-ov 7422	. . . . . . . 8 ⊢ (𝑚𝐺𝑚) = (𝐺‘⟨𝑚, 𝑚⟩)
10	8, 9	eqtr4di 2818	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑚))
11		fveq2 6885	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑚⟩))
12		df-ov 7422	. . . . . . . 8 ⊢ (𝑚𝐻𝑚) = (𝐻‘⟨𝑚, 𝑚⟩)
13	11, 12	eqtr4di 2818	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑚))
14	10, 13	imaeq12d 6065	. . . . . 6 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
15	14	eleq2d 2851	. . . . 5 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)) ↔ (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚))))
16		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (◡𝐹 “ {𝑋}))
17	16, 16	opelxpd 5702	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ⟨𝑚, 𝑚⟩ ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋})))
18		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
19		eqid 2765	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
20		imaid.i	. . . . . . . 8 ⊢ 𝐼 = (Id‘𝐸)
21		imassc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐹(𝐷 Func 𝐸)𝐺)
23		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘𝐸)
24	18, 23, 21	funcf1 17947	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
25	24	ffnd 6710	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐷))
26		fniniseg 7059	. . . . . . . . . . 11 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
28	27	biimpa 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋))
29	28	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (Base‘𝐷))
30	18, 19, 20, 22, 29	funcid 17951	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘(𝐹‘𝑚)))
31	28	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐹‘𝑚) = 𝑋)
32	31	fveq2d 6889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘(𝐹‘𝑚)) = (𝐼‘𝑋))
33	30, 32	eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘𝑋))
34		imasubc.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐷)
35	22	funcrcl2 49916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐷 ∈ Cat)
36	18, 34, 19, 35, 29	catidcl 17762	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚))
37		eqid 2765	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	18, 34, 37, 22, 29, 29	funcf2 17949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚𝐺𝑚):(𝑚𝐻𝑚)⟶((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑚)))
39	38	funfvima2d 7237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) ∧ ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚)) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
40	36, 39	mpdan 700	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
41	33, 40	eqeltrrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
42	15, 17, 41	rspcedvdw 3586	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
43	7, 42	exlimddv 1968	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
44	43	eliund 4965	. 2 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
45		relfunc 17943	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
46	45	brrelex1i 5719	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
47	21, 46	syl 18	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
48		imasubc.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
49	47, 47, 1, 1, 48	imasubclem3 49943	. 2 ⊢ (𝜑 → (𝑋𝐾𝑋) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
50	44, 49	eleqtrrd 2868	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  Vcvv 3457  ∅c0 4286  {csn 4591  ⟨cop 4597  ∪ ciun 4958   class class class wbr 5111   × cxp 5661  ^◡ccnv 5662   “ cima 5666   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Basecbs 17293  Hom chom 17345  Idccid 17745   Func cfunc 17935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-ixp 8902  df-cat 17748  df-cid 17749  df-func 17939

This theorem is used by:  imasubc3  49993