Theorem funcid 17926

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	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
funcid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
funcid	⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))

Proof of Theorem funcid

Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
2	1, 1	oveq12d 7428	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
4	2, 3	fveq12d 6888	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = ((𝑋𝐺𝑋)‘( 1 ‘𝑋)))
5		2fveq3 6886	. . 3 ⊢ (𝑥 = 𝑋 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑋)))
6	4, 5	eqeq12d 2777	. 2 ⊢ (𝑥 = 𝑋 → (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ↔ ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋))))
7		funcid.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8		funcid.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9		eqid 2761	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
10		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12		funcid.1	. . . . . 6 ⊢ 1 = (Id‘𝐷)
13		funcid.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐸)
14		eqid 2761	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
15		eqid 2761	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
16		df-br 5109	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
17	7, 16	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18		funcrcl 17919	. . . . . . . 8 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
20	19	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	19	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
22	8, 9, 10, 11, 12, 13, 14, 15, 20, 21	isfunc 17920	. . . . 5 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
23	7, 22	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
24	23	simp3d 1160	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
25		simpl 487	. . . 4 ⊢ ((((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
26	25	ralimi 3100	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))) → ∀𝑥 ∈ 𝐵 ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
27	24, 26	syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
28		funcid.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
29	6, 27, 28	rspcdva 3581	1 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ⟨cop 4594   class class class wbr 5108   × cxp 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984   ↑_m cmap 8823  Xcixp 8894  Basecbs 17268  Hom chom 17320  compcco 17321  Catccat 17719  Idccid 17720   Func cfunc 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8825  df-ixp 8895  df-func 17914

This theorem is referenced by:  funcsect  17928  funcoppc  17931  cofucl  17944  funcres  17952  fthsect  17983  catcisolem  18166  prfcl  18258  evlfcl  18277  curf1cl  18283  curfcl  18287  curfuncf  18293  yonedainv  18336  imaid  49877  upciclem3  49891  fucoid  50071  fucorid  50085  precofvallem  50089  termcfuncval  50255