Theorem idfuval 17591

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 6789	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4		fveq2 6774	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5		idfuval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2796	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7		simpr 485	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
8	7	reseq2d 5891	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
9	7	sqxpeqd 5621	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
10		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
11	10	fveq2d 6778	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
12		idfuval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
13	11, 12	eqtr4di 2796	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
14	13	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻‘𝑧))
15	14	reseq2d 5891	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻‘𝑧)))
16	9, 15	mpteq12dv 5165	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))
17	8, 16	opeq12d 4812	. . . . 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 3872	. . . 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 17574	. . . 4 ⊢ idfunc = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
20		opex 5379	. . . 4 ⊢ ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩ ∈ V
21	18, 19, 20	fvmpt 6875	. . 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 2790	1 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⦋csb 3832  ⟨cop 4567   ↦ cmpt 5157   I cid 5488   × cxp 5587   ↾ cres 5591  ‘cfv 6433  Basecbs 16912  Hom chom 16973  Catccat 17373  id_funccidfu 17570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-idfu 17574

This theorem is referenced by:  idfu2nd  17592  idfu1st  17594  idfucl  17596  catcisolem  17825  curf2ndf  17965  idfusubc0  45423