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Theorem funcid 17199
 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 7168 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3 fveq2 6658 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
42, 3fveq12d 6665 . . 3 (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1𝑥)) = ((𝑋𝐺𝑋)‘( 1𝑋)))
5 2fveq3 6663 . . 3 (𝑥 = 𝑋 → (𝐼‘(𝐹𝑥)) = (𝐼‘(𝐹𝑋)))
64, 5eqeq12d 2774 . 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 2758 . . . . . 6 (Base‘𝐸) = (Base‘𝐸)
10 eqid 2758 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
11 eqid 2758 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
12 funcid.1 . . . . . 6 1 = (Id‘𝐷)
13 funcid.i . . . . . 6 𝐼 = (Id‘𝐸)
14 eqid 2758 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
15 eqid 2758 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
16 df-br 5033 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
177, 16sylib 221 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18 funcrcl 17192 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1917, 18syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
2019simpld 498 . . . . . 6 (𝜑𝐷 ∈ Cat)
2119simprd 499 . . . . . 6 (𝜑𝐸 ∈ Cat)
228, 9, 10, 11, 12, 13, 14, 15, 20, 21isfunc 17193 . . . . 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 235 . . . 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 1141 . . 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 486 . . . 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 3092 . . 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 3543 1 (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟨cop 4528   class class class wbr 5032   × cxp 5522  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692   ↑m cmap 8416  Xcixp 8479  Basecbs 16541  Hom chom 16634  compcco 16635  Catccat 16993  Idccid 16994   Func cfunc 17183 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-ixp 8480  df-func 17187 This theorem is referenced by:  funcsect  17201  funcoppc  17204  cofucl  17217  funcres  17225  fthsect  17254  catcisolem  17432  prfcl  17519  evlfcl  17538  curf1cl  17544  curfcl  17548  curfuncf  17554  yonedainv  17597
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