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Theorem funcid 17916
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 7450 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3 fveq2 6905 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
42, 3fveq12d 6912 . . 3 (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1𝑥)) = ((𝑋𝐺𝑋)‘( 1𝑋)))
5 2fveq3 6910 . . 3 (𝑥 = 𝑋 → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑋)))
64, 5eqeq12d 2752 . 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 2736 . . . . . 6 (Base‘𝐸) = (Base‘𝐸)
10 eqid 2736 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
11 eqid 2736 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
12 funcid.1 . . . . . 6 1 = (Id‘𝐷)
13 funcid.i . . . . . 6 𝐼 = (Id‘𝐸)
14 eqid 2736 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
15 eqid 2736 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
16 df-br 5143 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
177, 16sylib 218 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18 funcrcl 17909 . . . . . . . 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 17910 . . . . 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 1144 . . 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 3082 . . 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 3622 1 (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  cop 4631   class class class wbr 5142   × cxp 5682  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  m cmap 8867  Xcixp 8938  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709   Func cfunc 17900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ixp 8939  df-func 17904
This theorem is referenced by:  funcsect  17918  funcoppc  17921  cofucl  17934  funcres  17942  fthsect  17973  catcisolem  18156  prfcl  18249  evlfcl  18268  curf1cl  18274  curfcl  18278  curfuncf  18284  yonedainv  18327  upciclem3  48943  fucoid  49066  fucorid  49080  precofvallem  49084  termcfuncval  49190
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