Theorem imaidfu2 49586

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 2736	. . . 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 2736	. . . 4 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu 49585	. . 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 2737	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
9	1, 2, 8	idfu1sta 49576	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
10	9	imaeq1d 6024	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
11		ssid 3944	. . . . . . . 8 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
12		resiima 6041	. . . . . . . 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 2787	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
15	14	sqxpeqd 5663	. . . . 5 ⊢ (𝜑 → (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))) = ((Base‘𝐷) × (Base‘𝐷)))
16	15	reseq2d 5944	. . . 4 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))))
17		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
18	4, 17	homffn 17659	. . . . 5 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
19		fnresdm 6617	. . . . 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 2787	. . 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 2797	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = 𝑆)
24		eqidd 2737	. . . 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 7442	. . 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 2779	. 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 2789	1 ⊢ (𝜑 → 𝐽 = 𝐾)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  {csn 4567  ∪ ciun 4933   I cid 5525   × cxp 5629  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231  Hom_f chomf 17632   Func cfunc 17821  id_funccidfu 17822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-homf 17636  df-func 17825  df-idfu 17826

This theorem is referenced by:  idsubc  49635  idfullsubc  49636