Theorem imaidfu2 49609

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 2739	. . . 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 2739	. . . 4 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu 49608	. . 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 2740	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
9	1, 2, 8	idfu1sta 49599	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
10	9	imaeq1d 6012	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
11		ssid 3937	. . . . . . . 8 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
12		resiima 6029	. . . . . . . 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 2790	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
15	14	sqxpeqd 5651	. . . . 5 ⊢ (𝜑 → (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))) = ((Base‘𝐷) × (Base‘𝐷)))
16	15	reseq2d 5932	. . . 4 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))))
17		eqid 2739	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
18	4, 17	homffn 17651	. . . . 5 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
19		fnresdm 6605	. . . . 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 2790	. . 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 2800	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = 𝑆)
24		eqidd 2740	. . . 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 7432	. . 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 2782	. 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 2792	1 ⊢ (𝜑 → 𝐽 = 𝐾)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  {csn 4556  ∪ ciun 4922   I cid 5513   × cxp 5617  ^◡ccnv 5618   ↾ cres 5621   “ cima 5622   Fn wfn 6481  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7930  2^nd c2nd 7931  Basecbs 17171  Hom chom 17223  Hom_f chomf 17624   Func cfunc 17813  id_funccidfu 17814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-map 8766  df-ixp 8837  df-cat 17626  df-cid 17627  df-homf 17628  df-func 17817  df-idfu 17818

This theorem is referenced by:  idsubc  49658  idfullsubc  49659