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Theorem idfuval 16359
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 fvex 6113 . . . . . 6 (Base‘𝑐) ∈ V
43a1i 11 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
5 fveq2 6103 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
6 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
75, 6syl6eqr 2662 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
8 simpr 476 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑏 = 𝐵)
98reseq2d 5317 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
108sqxpeqd 5065 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simpl 472 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6107 . . . . . . . . . 10 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 idfuval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1412, 13syl6eqr 2662 . . . . . . . . 9 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1514fveq1d 6105 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻𝑧))
1615reseq2d 5317 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻𝑧)))
1710, 16mpteq12dv 4663 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
189, 17opeq12d 4348 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
194, 7, 18csbied2 3527 . . . 4 (𝑐 = 𝐶(Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
20 df-idfu 16342 . . . 4 idfunc = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
21 opex 4859 . . . 4 ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩ ∈ V
2219, 20, 21fvmpt 6191 . . 3 (𝐶 ∈ Cat → (idfunc𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
232, 22syl 17 . 2 (𝜑 → (idfunc𝐶) = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
241, 23syl5eq 2656 1 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  csb 3499  cop 4131  cmpt 4643   I cid 4948   × cxp 5036  cres 5040  cfv 5804  Basecbs 15695  Hom chom 15779  Catccat 16148  idfunccidfu 16338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-idfu 16342
This theorem is referenced by:  idfu2nd  16360  idfu1st  16362  idfucl  16364  catcisolem  16579  curf2ndf  16710  idfusubc0  41655
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