Theorem imaid 49629

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 2846	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
4		inisegn0a 49311	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑋}) ≠ ∅)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ ∅)
6		n0 4293	. . . . 5 ⊢ ((◡𝐹 “ {𝑋}) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
7	5, 6	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
8		fveq2 6840	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑚⟩))
9		df-ov 7370	. . . . . . . 8 ⊢ (𝑚𝐺𝑚) = (𝐺‘⟨𝑚, 𝑚⟩)
10	8, 9	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑚))
11		fveq2 6840	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑚⟩))
12		df-ov 7370	. . . . . . . 8 ⊢ (𝑚𝐻𝑚) = (𝐻‘⟨𝑚, 𝑚⟩)
13	11, 12	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑚))
14	10, 13	imaeq12d 6026	. . . . . 6 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
15	14	eleq2d 2822	. . . . 5 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)) ↔ (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚))))
16		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (◡𝐹 “ {𝑋}))
17	16, 16	opelxpd 5670	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ⟨𝑚, 𝑚⟩ ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋})))
18		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
19		eqid 2736	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐹(𝐷 Func 𝐸)𝐺)
23		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘𝐸)
24	18, 23, 21	funcf1 17833	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
25	24	ffnd 6669	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐷))
26		fniniseg 7012	. . . . . . . . . . 11 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
28	27	biimpa 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋))
29	28	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (Base‘𝐷))
30	18, 19, 20, 22, 29	funcid 17837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘(𝐹‘𝑚)))
31	28	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐹‘𝑚) = 𝑋)
32	31	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘(𝐹‘𝑚)) = (𝐼‘𝑋))
33	30, 32	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘𝑋))
34		imasubc.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐷)
35	22	funcrcl2 49554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐷 ∈ Cat)
36	18, 34, 19, 35, 29	catidcl 17648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚))
37		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	18, 34, 37, 22, 29, 29	funcf2 17835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚𝐺𝑚):(𝑚𝐻𝑚)⟶((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑚)))
39	38	funfvima2d 7187	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) ∧ ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚)) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
40	36, 39	mpdan 688	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
41	33, 40	eqeltrrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
42	15, 17, 41	rspcedvdw 3567	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
43	7, 42	exlimddv 1937	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
44	43	eliund 4940	. 2 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
45		relfunc 17829	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
46	45	brrelex1i 5687	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
47	21, 46	syl 17	. . 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 49581	. 2 ⊢ (𝜑 → (𝑋𝐾𝑋) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
50	44, 49	eleqtrrd 2839	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429  ∅c0 4273  {csn 4567  ⟨cop 4573  ∪ ciun 4933   class class class wbr 5085   × cxp 5629  ^◡ccnv 5630   “ cima 5634   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Hom chom 17231  Idccid 17631   Func cfunc 17821

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-func 17825

This theorem is referenced by:  imasubc3  49631