Theorem idfuval 17922

Description: Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
idfuval.i	⊢ 𝐼 = (idfunc‘𝐶)
idfuval.b	⊢ 𝐵 = (Base‘𝐶)
idfuval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idfuval.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
idfuval	⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)

Distinct variable groups:   𝑧,𝐵   𝑧,𝐶   𝑧,𝐻   𝜑,𝑧

Allowed substitution hint:   𝐼(𝑧)

Proof of Theorem idfuval

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfuval.i	. 2 ⊢ 𝐼 = (idfunc‘𝐶)
2		idfuval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fvexd 6920	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4		fveq2 6905	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5		idfuval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2794	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7		simpr 484	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
8	7	reseq2d 5996	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
9	7	sqxpeqd 5716	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
10		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
11	10	fveq2d 6909	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
12		idfuval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
13	11, 12	eqtr4di 2794	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
14	13	fveq1d 6907	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻‘𝑧))
15	14	reseq2d 5996	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻‘𝑧)))
16	9, 15	mpteq12dv 5232	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
17	8, 16	opeq12d 4880	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
18	3, 6, 17	csbied2 3935	. . . 4 ⊢ (𝑐 = 𝐶 → ⦋(Base‘𝑐) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
19		df-idfu 17905	. . . 4 ⊢ idfunc = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
20		opex 5468	. . . 4 ⊢ ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩ ∈ V
21	18, 19, 20	fvmpt 7015	. . 3 ⊢ (𝐶 ∈ Cat → (idfunc‘𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
22	2, 21	syl 17	. 2 ⊢ (𝜑 → (idfunc‘𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
23	1, 22	eqtrid 2788	1 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3479  ⦋csb 3898  ⟨cop 4631   ↦ cmpt 5224   I cid 5576   × cxp 5682   ↾ cres 5686  ‘cfv 6560  Basecbs 17248  Hom chom 17309  Catccat 17708  id_funccidfu 17901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-idfu 17905

This theorem is referenced by:  idfu2nd  17923  idfu1st  17925  idfucl  17927  idfusubc0  17945  catcisolem  18156  curf2ndf  18293  idfudiag1bas  49182  idfudiag1  49183