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Theorem idfuval 16729
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 6356 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6344 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2804 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 simpr 479 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑏 = 𝐵)
87reseq2d 5543 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
97sqxpeqd 5290 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
10 simpl 474 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
1110fveq2d 6348 . . . . . . . . . 10 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
12 idfuval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1311, 12syl6eqr 2804 . . . . . . . . 9 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1413fveq1d 6346 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻𝑧))
1514reseq2d 5543 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻𝑧)))
169, 15mpteq12dv 4877 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
178, 16opeq12d 4553 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
183, 6, 17csbied2 3694 . . . 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 16712 . . . 4 idfunc = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
20 opex 5073 . . . 4 ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩ ∈ V
2118, 19, 20fvmpt 6436 . . 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 2798 1 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  csb 3666  cop 4319  cmpt 4873   I cid 5165   × cxp 5256  cres 5260  cfv 6041  Basecbs 16051  Hom chom 16146  Catccat 16518  idfunccidfu 16708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-idfu 16712
This theorem is referenced by:  idfu2nd  16730  idfu1st  16732  idfucl  16734  catcisolem  16949  curf2ndf  17080  idfusubc0  42367
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