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

Theorem funcid 16301
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 funcid.x . 2 (𝜑𝑋𝐵)
2 funcid.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
3 funcid.b . . . . . 6 𝐵 = (Base‘𝐷)
4 eqid 2609 . . . . . 6 (Base‘𝐸) = (Base‘𝐸)
5 eqid 2609 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
6 eqid 2609 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
7 funcid.1 . . . . . 6 1 = (Id‘𝐷)
8 funcid.i . . . . . 6 𝐼 = (Id‘𝐸)
9 eqid 2609 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
10 eqid 2609 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
11 df-br 4578 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
122, 11sylib 206 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
13 funcrcl 16294 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1412, 13syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1514simpld 473 . . . . . 6 (𝜑𝐷 ∈ Cat)
1614simprd 477 . . . . . 6 (𝜑𝐸 ∈ Cat)
173, 4, 5, 6, 7, 8, 9, 10, 15, 16isfunc 16295 . . . . 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))))))
182, 17mpbid 220 . . . 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)))))
1918simp3d 1067 . . 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))))
20 simpl 471 . . . 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)))
2120ralimi 2935 . . 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)))
2219, 21syl 17 . 2 (𝜑 → ∀𝑥𝐵 ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
23 id 22 . . . . . 6 (𝑥 = 𝑋𝑥 = 𝑋)
2423, 23oveq12d 6544 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
25 fveq2 6087 . . . . 5 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
2624, 25fveq12d 6093 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1𝑥)) = ((𝑋𝐺𝑋)‘( 1𝑋)))
27 fveq2 6087 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2827fveq2d 6091 . . . 4 (𝑥 = 𝑋 → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑋)))
2926, 28eqeq12d 2624 . . 3 (𝑥 = 𝑋 → (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ↔ ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋))))
3029rspcv 3277 . 2 (𝑋𝐵 → (∀𝑥𝐵 ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋))))
311, 22, 30sylc 62 1 (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cop 4130   class class class wbr 4577   × cxp 5025  wf 5785  cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  Xcixp 7771  Basecbs 15643  Hom chom 15727  compcco 15728  Catccat 16096  Idccid 16097   Func cfunc 16285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-map 7723  df-ixp 7772  df-func 16289
This theorem is referenced by:  funcsect  16303  funcoppc  16306  cofucl  16319  funcres  16327  fthsect  16356  catcisolem  16527  prfcl  16614  evlfcl  16633  curf1cl  16639  curfcl  16643  curfuncf  16649  yonedainv  16692
  Copyright terms: Public domain W3C validator