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Theorem funcid 16737
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 6811 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3 fveq2 6332 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
42, 3fveq12d 6338 . . 3 (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1𝑥)) = ((𝑋𝐺𝑋)‘( 1𝑋)))
5 fveq2 6332 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
65fveq2d 6336 . . 3 (𝑥 = 𝑋 → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑋)))
74, 6eqeq12d 2786 . 2 (𝑥 = 𝑋 → (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ↔ ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋))))
8 funcid.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
9 funcid.b . . . . . 6 𝐵 = (Base‘𝐷)
10 eqid 2771 . . . . . 6 (Base‘𝐸) = (Base‘𝐸)
11 eqid 2771 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
12 eqid 2771 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
13 funcid.1 . . . . . 6 1 = (Id‘𝐷)
14 funcid.i . . . . . 6 𝐼 = (Id‘𝐸)
15 eqid 2771 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
16 eqid 2771 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
17 df-br 4787 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
188, 17sylib 208 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
19 funcrcl 16730 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
2018, 19syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
2120simpld 482 . . . . . 6 (𝜑𝐷 ∈ Cat)
2220simprd 483 . . . . . 6 (𝜑𝐸 ∈ Cat)
239, 10, 11, 12, 13, 14, 15, 16, 21, 22isfunc 16731 . . . . 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))))))
248, 23mpbid 222 . . . 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)))))
2524simp3d 1138 . . 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))))
26 simpl 468 . . . 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)))
2726ralimi 3101 . . 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)))
2825, 27syl 17 . 2 (𝜑 → ∀𝑥𝐵 ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
29 funcid.x . 2 (𝜑𝑋𝐵)
307, 28, 29rspcdva 3466 1 (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cop 4322   class class class wbr 4786   × cxp 5247  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  𝑚 cmap 8009  Xcixp 8062  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533   Func cfunc 16721
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-ixp 8063  df-func 16725
This theorem is referenced by:  funcsect  16739  funcoppc  16742  cofucl  16755  funcres  16763  fthsect  16792  catcisolem  16963  prfcl  17051  evlfcl  17070  curf1cl  17076  curfcl  17080  curfuncf  17086  yonedainv  17129
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