Theorem imaidfu 49017

Description: The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)

Hypotheses

Ref	Expression
imaidfu.i	⊢ 𝐼 = (idfunc‘𝐶)
imaidfu.d	⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
imaidfu.h	⊢ 𝐻 = (Hom ‘𝐷)
imaidfu.j	⊢ 𝐽 = (Homf ‘𝐷)
imaidfu.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))
imaidfu.s	⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴)

Assertion

Ref	Expression
imaidfu	⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)

Distinct variable groups:   𝐻,𝑝,𝑥,𝑦   𝐼,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦

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

Proof of Theorem imaidfu

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

Step	Hyp	Ref	Expression
1		imaidfu.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (idfunc‘𝐶)
2		imaidfu.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
3		eqidd 2736	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49008	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
6	5	cnveqd 5855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
7		cnvresid 6614	. . . . . . . . . 10 ⊢ ◡( I ↾ (Base‘𝐷)) = ( I ↾ (Base‘𝐷))
8	6, 7	eqtrdi 2786	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
9	8	fveq1d 6877	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = (( I ↾ (Base‘𝐷))‘𝑧))
10		imaidfu.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴)
11		imassrn 6058	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐼) “ 𝐴) ⊆ ran (1st ‘𝐼)
12	10, 11	eqsstri 4005	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ran (1st ‘𝐼)
13	4	rneqd 5918	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (1st ‘𝐼) = ran ( I ↾ (Base‘𝐷)))
14		rnresi 6062	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (Base‘𝐷)) = (Base‘𝐷)
15	13, 14	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (1st ‘𝐼) = (Base‘𝐷))
16	12, 15	sseqtrid 4001	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐷))
18		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
19	17, 18	sseldd 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐷))
20		fvresi 7164	. . . . . . . . 9 ⊢ (𝑧 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑧) = 𝑧)
21	19, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (( I ↾ (Base‘𝐷))‘𝑧) = 𝑧)
22	9, 21	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = 𝑧)
23	8	fveq1d 6877	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑤) = (( I ↾ (Base‘𝐷))‘𝑤))
24		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ 𝑆)
25	17, 24	sseldd 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ (Base‘𝐷))
26		fvresi 7164	. . . . . . . . 9 ⊢ (𝑤 ∈ (Base‘𝐷) → (( I ↾ (Base‘𝐷))‘𝑤) = 𝑤)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (( I ↾ (Base‘𝐷))‘𝑤) = 𝑤)
28	23, 27	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑤) = 𝑤)
29	22, 28	oveq12d 7421	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)(2nd ‘𝐼)(◡(1st ‘𝐼)‘𝑤)) = (𝑧(2nd ‘𝐼)𝑤))
30	22, 28	oveq12d 7421	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)𝐻(◡(1st ‘𝐼)‘𝑤)) = (𝑧𝐻𝑤))
31	29, 30	imaeq12d 6048	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (((◡(1st ‘𝐼)‘𝑧)(2nd ‘𝐼)(◡(1st ‘𝐼)‘𝑤)) “ ((◡(1st ‘𝐼)‘𝑧)𝐻(◡(1st ‘𝐼)‘𝑤))) = ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)))
32		f1oi 6855	. . . . . . . 8 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
33	5	f1oeq1d 6812	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((1st ‘𝐼):(Base‘𝐷)–1-1-onto→(Base‘𝐷) ↔ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)))
34	32, 33	mpbiri 258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼):(Base‘𝐷)–1-1-onto→(Base‘𝐷))
35		f1of1 6816	. . . . . . 7 ⊢ ((1st ‘𝐼):(Base‘𝐷)–1-1-onto→(Base‘𝐷) → (1st ‘𝐼):(Base‘𝐷)–1-1→(Base‘𝐷))
36	34, 35	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼):(Base‘𝐷)–1-1→(Base‘𝐷))
37		fvexd 6890	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼) ∈ V)
38		imaidfu.k	. . . . . 6 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))
39	10, 36, 18, 24, 37, 38	imaf1hom 49015	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐾𝑤) = (((◡(1st ‘𝐼)‘𝑧)(2nd ‘𝐼)(◡(1st ‘𝐼)‘𝑤)) “ ((◡(1st ‘𝐼)‘𝑧)𝐻(◡(1st ‘𝐼)‘𝑤))))
40		imaidfu.j	. . . . . . 7 ⊢ 𝐽 = (Homf ‘𝐷)
41		eqid 2735	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
42		imaidfu.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
43	40, 41, 42, 19, 25	homfval 17702	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐻𝑤))
44	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐼 ∈ (𝐷 Func 𝐸))
45		eqidd 2736	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (Base‘𝐷) = (Base‘𝐷))
46	42	oveqi 7416	. . . . . . . . . 10 ⊢ (𝑧𝐻𝑤) = (𝑧(Hom ‘𝐷)𝑤)
47	46	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐻𝑤) = (𝑧(Hom ‘𝐷)𝑤))
48	1, 44, 45, 19, 25, 47	idfu2nda 49010	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧(2nd ‘𝐼)𝑤) = ( I ↾ (𝑧𝐻𝑤)))
49	48	imaeq1d 6046	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)))
50		ssid 3981	. . . . . . . 8 ⊢ (𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤)
51		resiima 6063	. . . . . . . 8 ⊢ ((𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤) → (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤)
53	49, 52	eqtrdi 2786	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
54	43, 53	eqtr4d 2773	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)))
55	31, 39, 54	3eqtr4rd 2781	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
56	55	ralrimivva 3187	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
57		fveq2 6875	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝐽‘⟨𝑧, 𝑤⟩))
58		df-ov 7406	. . . . . 6 ⊢ (𝑧𝐽𝑤) = (𝐽‘⟨𝑧, 𝑤⟩)
59	57, 58	eqtr4di 2788	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝑧𝐽𝑤))
60		fveq2 6875	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝐾‘⟨𝑧, 𝑤⟩))
61		df-ov 7406	. . . . . 6 ⊢ (𝑧𝐾𝑤) = (𝐾‘⟨𝑧, 𝑤⟩)
62	60, 61	eqtr4di 2788	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝑧𝐾𝑤))
63	59, 62	eqeq12d 2751	. . . 4 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝐽‘𝑞) = (𝐾‘𝑞) ↔ (𝑧𝐽𝑤) = (𝑧𝐾𝑤)))
64	63	ralxp 5821	. . 3 ⊢ (∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
65	56, 64	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞))
66	40, 41	homffn 17703	. . . 4 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
67	66	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷)))
68		fvexd 6890	. . . 4 ⊢ (𝜑 → (1st ‘𝐼) ∈ V)
69	68, 68, 38	imasubclem2 49012	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
70		xpss12 5669	. . . 4 ⊢ ((𝑆 ⊆ (Base‘𝐷) ∧ 𝑆 ⊆ (Base‘𝐷)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
71	16, 16, 70	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
72		fvreseq1 7028	. . 3 ⊢ (((𝐽 Fn ((Base‘𝐷) × (Base‘𝐷)) ∧ 𝐾 Fn (𝑆 × 𝑆)) ∧ (𝑆 × 𝑆) ⊆ ((Base‘𝐷) × (Base‘𝐷))) → ((𝐽 ↾ (𝑆 × 𝑆)) = 𝐾 ↔ ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞)))
73	67, 69, 71, 72	syl21anc 837	. 2 ⊢ (𝜑 → ((𝐽 ↾ (𝑆 × 𝑆)) = 𝐾 ↔ ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞)))
74	65, 73	mpbird 257	1 ⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ⊆ wss 3926  {csn 4601  ⟨cop 4607  ∪ ciun 4967   I cid 5547   × cxp 5652  ^◡ccnv 5653  ran crn 5655   ↾ cres 5656   “ cima 5657   Fn wfn 6525  –1-1→wf1 6527  –1-1-onto→wf1o 6529  ‘cfv 6530  (class class class)co 7403   ∈ cmpo 7405  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17226  Hom chom 17280  Hom_f chomf 17676   Func cfunc 17865  id_funccidfu 17866

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-map 8840  df-ixp 8910  df-cat 17678  df-cid 17679  df-homf 17680  df-func 17869  df-idfu 17870

This theorem is referenced by:  imaidfu2  49018