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

Theorem funcid 17921
Description: A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcid.b 𝐵 = (Base‘𝐷)
funcid.1 1 = (Id‘𝐷)
funcid.i 𝐼 = (Id‘𝐸)
funcid.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funcid.x (𝜑𝑋𝐵)
Assertion
Ref Expression
funcid (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))

Proof of Theorem funcid
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
21, 1oveq12d 7449 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3 fveq2 6907 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
42, 3fveq12d 6914 . . 3 (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1𝑥)) = ((𝑋𝐺𝑋)‘( 1𝑋)))
5 2fveq3 6912 . . 3 (𝑥 = 𝑋 → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑋)))
64, 5eqeq12d 2751 . 2 (𝑥 = 𝑋 → (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ↔ ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋))))
7 funcid.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
8 funcid.b . . . . . 6 𝐵 = (Base‘𝐷)
9 eqid 2735 . . . . . 6 (Base‘𝐸) = (Base‘𝐸)
10 eqid 2735 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
11 eqid 2735 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
12 funcid.1 . . . . . 6 1 = (Id‘𝐷)
13 funcid.i . . . . . 6 𝐼 = (Id‘𝐸)
14 eqid 2735 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
15 eqid 2735 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
16 df-br 5149 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
177, 16sylib 218 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18 funcrcl 17914 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1917, 18syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
2019simpld 494 . . . . . 6 (𝜑𝐷 ∈ Cat)
2119simprd 495 . . . . . 6 (𝜑𝐸 ∈ Cat)
228, 9, 10, 11, 12, 13, 14, 15, 20, 21isfunc 17915 . . . . 5 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))(Hom ‘𝐸)(𝐹‘(2nd𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
237, 22mpbid 232 . . . 4 (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))(Hom ‘𝐸)(𝐹‘(2nd𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
2423simp3d 1143 . . 3 (𝜑 → ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
25 simpl 482 . . . 4 ((((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
2625ralimi 3081 . . 3 (∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))) → ∀𝑥𝐵 ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
2724, 26syl 17 . 2 (𝜑 → ∀𝑥𝐵 ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
28 funcid.x . 2 (𝜑𝑋𝐵)
296, 27, 28rspcdva 3623 1 (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cop 4637   class class class wbr 5148   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  Xcixp 8936  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710   Func cfunc 17905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ixp 8937  df-func 17909
This theorem is referenced by:  funcsect  17923  funcoppc  17926  cofucl  17939  funcres  17947  fthsect  17979  catcisolem  18164  prfcl  18259  evlfcl  18279  curf1cl  18285  curfcl  18289  curfuncf  18295  yonedainv  18338  upciclem3  48814
  Copyright terms: Public domain W3C validator