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Theorem idfuval 16743
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 6346 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6333 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2823 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 simpr 471 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑏 = 𝐵)
87reseq2d 5533 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵))
97sqxpeqd 5281 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
10 simpl 468 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
1110fveq2d 6337 . . . . . . . . . 10 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
12 idfuval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1311, 12syl6eqr 2823 . . . . . . . . 9 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1413fveq1d 6335 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻𝑧))
1514reseq2d 5533 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻𝑧)))
169, 15mpteq12dv 4868 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
178, 16opeq12d 4548 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
183, 6, 17csbied2 3710 . . . 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 16726 . . . 4 idfunc = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))⟩)
20 opex 5061 . . . 4 ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩ ∈ V
2118, 19, 20fvmpt 6426 . . 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 2817 1 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  cop 4323  cmpt 4864   I cid 5157   × cxp 5248  cres 5252  cfv 6030  Basecbs 16064  Hom chom 16160  Catccat 16532  idfunccidfu 16722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-idfu 16726
This theorem is referenced by:  idfu2nd  16744  idfu1st  16746  idfucl  16748  catcisolem  16963  curf2ndf  17095  idfusubc0  42390
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