Theorem imaidfu2 49917

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 2763	. . . 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 2763	. . . 4 ⊢ ((1st ‘𝐼) “ (Base‘𝐷)) = ((1st ‘𝐼) “ (Base‘𝐷))
7	1, 2, 3, 4, 5, 6	imaidfu 49916	. . 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 2764	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
9	1, 2, 8	idfu1sta 49907	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
10	9	imaeq1d 6061	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
11		ssid 3959	. . . . . . . 8 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
12		resiima 6078	. . . . . . . 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 2814	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
15	14	sqxpeqd 5693	. . . . 5 ⊢ (𝜑 → (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷))) = ((Base‘𝐷) × (Base‘𝐷)))
16	15	reseq2d 5978	. . . 4 ⊢ (𝜑 → (𝐽 ↾ (((1st ‘𝐼) “ (Base‘𝐷)) × ((1st ‘𝐼) “ (Base‘𝐷)))) = (𝐽 ↾ ((Base‘𝐷) × (Base‘𝐷))))
17		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
18	4, 17	homffn 17753	. . . . 5 ⊢ 𝐽 Fn ((Base‘𝐷) × (Base‘𝐷))
19		fnresdm 6654	. . . . 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 2814	. . 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 2824	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = 𝑆)
24		eqidd 2764	. . . 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 7485	. . 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 2806	. 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 2816	1 ⊢ (𝜑 → 𝐽 = 𝐾)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ∪ ciun 4956   I cid 5555   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17273  Hom chom 17325  Hom_f chomf 17726   Func cfunc 17915  id_funccidfu 17916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-cat 17728  df-cid 17729  df-homf 17730  df-func 17919  df-idfu 17920

This theorem is used by:  idsubc  49966  idfullsubc  49967