Theorem imaidfu2 49737

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)))
imaidfu2.s	⊢ (𝜑 → 𝑆 = (Base‘𝐷))

Assertion

Ref	Expression
imaidfu2	⊢ (𝜑 → 𝐽 = 𝐾)

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

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

Proof of Theorem imaidfu2

Step	Hyp	Ref	Expression
1		imaidfu.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
2		imaidfu.d	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
3		imaidfu.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
4		imaidfu.j	. . . 4 ⊢ 𝐽 = (Homf ‘𝐷)
5		eqid 2764	. . . 4 ⊢ (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))) = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))
6		eqid 2764	. . . 4 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu 49736	. . 3 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))))
8		eqidd 2765	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
9	1, 2, 8	idfu1sta 49727	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
10	9	imaeq1d 6050	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
11		ssid 3960	. . . . . . . 8 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
12		resiima 6067	. . . . . . . 8 ⊢ ((Base‘𝐷) ⊆ (Base‘𝐷) → (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)) = (Base‘𝐷))
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)) = (Base‘𝐷)
14	10, 13	eqtrdi 2815	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
15	14	sqxpeqd 5681	. . . . 5 ⊢ (𝜑 → (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))) = ((Base‘𝐷) × (Base‘𝐷)))
16	15	reseq2d 5967	. . . 4 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))))
17		eqid 2764	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
18	4, 17	homffn 17727	. . . . 5 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
19		fnresdm 6642	. . . . 5 ⊢ (𝐽 Fn ((Base‘𝐷) × (Base‘𝐷)) → (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))) = 𝐽)
20	18, 19	ax-mp 5	. . . 4 ⊢ (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))) = 𝐽
21	16, 20	eqtrdi 2815	. . 3 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = 𝐽)
22		imaidfu2.s	. . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
23	13, 10, 22	3eqtr4a 2825	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = 𝑆)
24		eqidd 2765	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))
25	23, 23, 24	mpoeq123dv 7473	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((1st ‘𝐼) “ (Base‘𝐷)), 𝑦 ∈ ((1st ‘𝐼) “ (Base‘𝐷)) ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))))
26	7, 21, 25	3eqtr3d 2807	. 2 ⊢ (𝜑 → 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))))
27		imaidfu.k	. 2 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))
28	26, 27	eqtr4di 2817	1 ⊢ (𝜑 → 𝐽 = 𝐾)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ⊆ wss 3906  {csn 4584  ∪ ciun 4951   I cid 5543   × cxp 5647  ^◡ccnv 5648   ↾ cres 5651   “ cima 5652   Fn wfn 6518  ‘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:  idsubc  49786  idfullsubc  49787