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Theorem idfuval 16301
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 6094 . . . . . 6 (Base‘𝑐) ∈ V
43a1i 11 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
5 fveq2 6084 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
6 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
75, 6syl6eqr 2657 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
8 simpr 475 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑏 = 𝐵)
98reseq2d 5300 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
108sqxpeqd 5051 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simpl 471 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6088 . . . . . . . . . 10 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 idfuval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1412, 13syl6eqr 2657 . . . . . . . . 9 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1514fveq1d 6086 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻𝑧))
1615reseq2d 5300 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻𝑧)))
1710, 16mpteq12dv 4653 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
189, 17opeq12d 4338 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
194, 7, 18csbied2 3522 . . . 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 16284 . . . 4 idfunc = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
21 opex 4849 . . . 4 ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩ ∈ V
2219, 20, 21fvmpt 6172 . . 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 2651 1 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  csb 3494  cop 4126  cmpt 4633   I cid 4934   × cxp 5022  cres 5026  cfv 5786  Basecbs 15637  Hom chom 15721  Catccat 16090  idfunccidfu 16280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-res 5036  df-iota 5750  df-fun 5788  df-fv 5794  df-idfu 16284
This theorem is referenced by:  idfu2nd  16302  idfu1st  16304  idfucl  16306  catcisolem  16521  curf2ndf  16652  idfusubc0  41653
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