Theorem imaidfu 49736

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 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49727	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
6	5	cnveqd 5849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ◡( I ↾ (Base‘𝐷)))
7		cnvresid 6602	. . . . . . . . . 10 ⊢ ◡( I ↾ (Base‘𝐷)) = ( I ↾ (Base‘𝐷))
8	6, 7	eqtrdi 2815	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ◡(1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
9	8	fveq1d 6871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = (( I ↾ (Base‘𝐷))‘𝑧))
10		imaidfu.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴)
11		imassrn 6062	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐼) “ 𝐴) ⊆ ran (1st ‘𝐼)
12	10, 11	eqsstri 3984	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ran (1st ‘𝐼)
13	4	rneqd 5916	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (1st ‘𝐼) = ran ( I ↾ (Base‘𝐷)))
14		rnresi 6066	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (Base‘𝐷)) = (Base‘𝐷)
15	13, 14	eqtrdi 2815	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (1st ‘𝐼) = (Base‘𝐷))
16	12, 15	sseqtrid 3980	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
17	16	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐷))
18		simprl 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
19	17, 18	sseldd 3939	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐷))
20		fvresi 7159	. . . . . . . . 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 2799	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑧) = 𝑧)
23	8	fveq1d 6871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑤) = (( I ↾ (Base‘𝐷))‘𝑤))
24		simprr 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ 𝑆)
25	17, 24	sseldd 3939	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝑤 ∈ (Base‘𝐷))
26		fvresi 7159	. . . . . . . . 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 2799	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (◡(1st ‘𝐼)‘𝑤) = 𝑤)
29	22, 28	oveq12d 7416	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)(2nd ‘𝐼)(◡(1st ‘𝐼)‘𝑤)) = (𝑧(2nd ‘𝐼)𝑤))
30	22, 28	oveq12d 7416	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((◡(1st ‘𝐼)‘𝑧)𝐻(◡(1st ‘𝐼)‘𝑤)) = (𝑧𝐻𝑤))
31	29, 30	imaeq12d 6052	. . . . 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 6847	. . . . . . . 8 ⊢ ( I ↾ (Base‘𝐷)):(Base‘𝐷)–1-1-onto→(Base‘𝐷)
33	5	f1oeq1d 6803	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (1st ‘𝐼):(Base‘𝐷)–1-1-onto→(Base‘𝐷))
35		f1of1 6807	. . . . . . 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 6884	. . . . . 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 49734	. . . . 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 2764	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
42		imaidfu.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
43	40, 41, 42, 19, 25	homfval 17726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐻𝑤))
44	2	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐼 ∈ (𝐷 Func 𝐸))
45		eqidd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (Base‘𝐷) = (Base‘𝐷))
46	42	oveqi 7411	. . . . . . . . . 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 49729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧(2nd ‘𝐼)𝑤) = ( I ↾ (𝑧𝐻𝑤)))
49	48	imaeq1d 6050	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)))
50		ssid 3960	. . . . . . . 8 ⊢ (𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤)
51		resiima 6067	. . . . . . . 8 ⊢ ((𝑧𝐻𝑤) ⊆ (𝑧𝐻𝑤) → (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ (( I ↾ (𝑧𝐻𝑤)) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤)
53	49, 52	eqtrdi 2815	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)) = (𝑧𝐻𝑤))
54	43, 53	eqtr4d 2802	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = ((𝑧(2nd ‘𝐼)𝑤) “ (𝑧𝐻𝑤)))
55	31, 39, 54	3eqtr4rd 2810	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
56	55	ralrimivva 3207	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
57		fveq2 6869	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝐽‘⟨𝑧, 𝑤⟩))
58		df-ov 7401	. . . . . 6 ⊢ (𝑧𝐽𝑤) = (𝐽‘⟨𝑧, 𝑤⟩)
59	57, 58	eqtr4di 2817	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐽‘𝑞) = (𝑧𝐽𝑤))
60		fveq2 6869	. . . . . 6 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝐾‘⟨𝑧, 𝑤⟩))
61		df-ov 7401	. . . . . 6 ⊢ (𝑧𝐾𝑤) = (𝐾‘⟨𝑧, 𝑤⟩)
62	60, 61	eqtr4di 2817	. . . . 5 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → (𝐾‘𝑞) = (𝑧𝐾𝑤))
63	59, 62	eqeq12d 2780	. . . 4 ⊢ (𝑞 = ⟨𝑧, 𝑤⟩ → ((𝐽‘𝑞) = (𝐾‘𝑞) ↔ (𝑧𝐽𝑤) = (𝑧𝐾𝑤)))
64	63	ralxp 5815	. . 3 ⊢ (∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (𝑧𝐽𝑤) = (𝑧𝐾𝑤))
65	56, 64	sylibr 236	. 2 ⊢ (𝜑 → ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞))
66	40, 41	homffn 17727	. . . 4 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
67	66	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷)))
68		fvexd 6884	. . . 4 ⊢ (𝜑 → (1st ‘𝐼) ∈ V)
69	68, 68, 38	imasubclem2 49731	. . 3 ⊢ (𝜑 → 𝐾 Fn (𝑆 × 𝑆))
70		xpss12 5664	. . . 4 ⊢ ((𝑆 ⊆ (Base‘𝐷) ∧ 𝑆 ⊆ (Base‘𝐷)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
71	16, 16, 70	syl2anc 593	. . 3 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
72		fvreseq1 7022	. . 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 848	. 2 ⊢ (𝜑 → ((𝐽 ↾ (𝑆 × 𝑆)) = 𝐾 ↔ ∀𝑞 ∈ (𝑆 × 𝑆)(𝐽‘𝑞) = (𝐾‘𝑞)))
74	65, 73	mpbird 259	1 ⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  Vcvv 3456   ⊆ wss 3906  {csn 4584  ⟨cop 4590  ∪ ciun 4951   I cid 5543   × cxp 5647  ^◡ccnv 5648  ran crn 5650   ↾ cres 5651   “ cima 5652   Fn wfn 6518  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  1^st c1st 7970  2^nd c2nd 7971  Basecbs 17247  Hom chom 17299  Hom_f chomf 17700   Func cfunc 17889  id_funccidfu 17890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-ixp 8882  df-cat 17702  df-cid 17703  df-homf 17704  df-func 17893  df-idfu 17894

This theorem is referenced by:  imaidfu2  49737