MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idfuval Structured version   Visualization version   GIF version

Theorem idfuval 17016
Description: Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
idfuval.i 𝐼 = (idfunc𝐶)
idfuval.b 𝐵 = (Base‘𝐶)
idfuval.c (𝜑𝐶 ∈ Cat)
idfuval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
idfuval (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
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.
StepHypRef Expression
1 idfuval.i . 2 𝐼 = (idfunc𝐶)
2 idfuval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fvexd 6511 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6496 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2826 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 simpr 477 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑏 = 𝐵)
87reseq2d 5692 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
97sqxpeqd 5435 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
10 simpl 475 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
1110fveq2d 6500 . . . . . . . . . 10 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
12 idfuval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1311, 12syl6eqr 2826 . . . . . . . . 9 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1413fveq1d 6498 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻𝑧))
1514reseq2d 5692 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻𝑧)))
169, 15mpteq12dv 5008 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
178, 16opeq12d 4681 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
183, 6, 17csbied2 3810 . . . 4 (𝑐 = 𝐶(Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
19 df-idfu 16999 . . . 4 idfunc = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
20 opex 5209 . . . 4 ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩ ∈ V
2118, 19, 20fvmpt 6593 . . 3 (𝐶 ∈ Cat → (idfunc𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
222, 21syl 17 . 2 (𝜑 → (idfunc𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
231, 22syl5eq 2820 1 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  Vcvv 3409  csb 3780  cop 4441  cmpt 5004   I cid 5307   × cxp 5401  cres 5405  cfv 6185  Basecbs 16337  Hom chom 16430  Catccat 16805  idfunccidfu 16995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-res 5415  df-iota 6149  df-fun 6187  df-fv 6193  df-idfu 16999
This theorem is referenced by:  idfu2nd  17017  idfu1st  17019  idfucl  17021  catcisolem  17236  curf2ndf  17367  idfusubc0  43525
  Copyright terms: Public domain W3C validator