Theorem imaidfu 49121

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 2731	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49112	. . . . . . . . . . . 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 5813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
7		cnvresid 6556	. . . . . . . . . 10 ⊢ ◡( I ↾ (Base‘𝐷)) = ( I ↾ (Base‘𝐷))
8	6, 7	eqtrdi 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
9	8	fveq1d 6819	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = (( I ↾ (Base‘𝐷))‘𝑧))
10		imaidfu.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴)
11		imassrn 6017	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐼) “ 𝐴) ⊆ ran (1st ‘𝐼)
12	10, 11	eqsstri 3979	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ran (1st ‘𝐼)
13	4	rneqd 5875	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (1st ‘𝐼) = ran ( I ↾ (Base‘𝐷)))
14		rnresi 6021	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (Base‘𝐷)) = (Base‘𝐷)
15	13, 14	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (1st ‘𝐼) = (Base‘𝐷))
16	12, 15	sseqtrid 3975	. . . . . . . . . . 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 3933	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐷))
20		fvresi 7102	. . . . . . . . 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 2765	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = 𝑧)
23	8	fveq1d 6819	. . . . . . . 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 3933	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ (Base‘𝐷))
26		fvresi 7102	. . . . . . . . 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 2765	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑤) = 𝑤)
29	22, 28	oveq12d 7359	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)(2nd ‘𝐼)(◡(1st ‘𝐼)‘𝑤)) = (𝑧(2nd ‘𝐼)𝑤))
30	22, 28	oveq12d 7359	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)𝐻(◡(1st ‘𝐼)‘𝑤)) = (𝑧𝐻𝑤))
31	29, 30	imaeq12d 6007	. . . . 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 6797	. . . . . . . 8 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
33	5	f1oeq1d 6754	. . . . . . . 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 6758	. . . . . . 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 6832	. . . . . 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 49119	. . . . 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 2730	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
42		imaidfu.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
43	40, 41, 42, 19, 25	homfval 17590	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐻𝑤))
44	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐼 ∈ (𝐷 Func 𝐸))
45		eqidd 2731	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (Base‘𝐷) = (Base‘𝐷))
46	42	oveqi 7354	. . . . . . . . . 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 49114	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧(2nd ‘𝐼)𝑤) = ( I ↾ (𝑧𝐻𝑤)))
49	48	imaeq1d 6005	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)))
50		ssid 3955	. . . . . . . 8 ⊢ (𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤)
51		resiima 6022	. . . . . . . 8 ⊢ ((𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤) → (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤)
53	49, 52	eqtrdi 2781	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
54	43, 53	eqtr4d 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)))
55	31, 39, 54	3eqtr4rd 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
56	55	ralrimivva 3173	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
57		fveq2 6817	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝐽‘⟨𝑧, 𝑤⟩))
58		df-ov 7344	. . . . . 6 ⊢ (𝑧𝐽𝑤) = (𝐽‘⟨𝑧, 𝑤⟩)
59	57, 58	eqtr4di 2783	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝑧𝐽𝑤))
60		fveq2 6817	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝐾‘⟨𝑧, 𝑤⟩))
61		df-ov 7344	. . . . . 6 ⊢ (𝑧𝐾𝑤) = (𝐾‘⟨𝑧, 𝑤⟩)
62	60, 61	eqtr4di 2783	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝑧𝐾𝑤))
63	59, 62	eqeq12d 2746	. . . 4 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝐽‘𝑞) = (𝐾‘𝑞) ↔ (𝑧𝐽𝑤) = (𝑧𝐾𝑤)))
64	63	ralxp 5779	. . 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 17591	. . . 4 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
67	66	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷)))
68		fvexd 6832	. . . 4 ⊢ (𝜑 → (1st ‘𝐼) ∈ V)
69	68, 68, 38	imasubclem2 49116	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
70		xpss12 5629	. . . 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 6967	. . 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 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ⊆ wss 3900  {csn 4574  ⟨cop 4580  ∪ ciun 4939   I cid 5508   × cxp 5612  ^◡ccnv 5613  ran crn 5615   ↾ cres 5616   “ cima 5617   Fn wfn 6472  –1-1→wf1 6474  –1-1-onto→wf1o 6476  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  2^nd c2nd 7915  Basecbs 17112  Hom chom 17164  Hom_f chomf 17564   Func cfunc 17753  id_funccidfu 17754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-ixp 8817  df-cat 17566  df-cid 17567  df-homf 17568  df-func 17757  df-idfu 17758

This theorem is referenced by:  imaidfu2  49122