Theorem funcid 17933

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 7430	. . . 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 2778	. 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 2762	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
10		eqid 2762	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11		eqid 2762	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12		funcid.1	. . . . . 6 ⊢ 1 = (Id‘𝐷)
13		funcid.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐸)
14		eqid 2762	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
15		eqid 2762	. . . . . 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 17926	. . . . . . . 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 17927	. . . . 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 1161	. . 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 3101	. . 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ⟨cop 4594   class class class wbr 5108   × cxp 5658  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983   ↑_m cmap 8822  Xcixp 8893  Basecbs 17275  Hom chom 17327  compcco 17328  Catccat 17726  Idccid 17727   Func cfunc 17917

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  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 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-ixp 8894  df-func 17921

This theorem is used by:  funcsect  17935  funcoppc  17938  cofucl  17951  funcres  17959  fthsect  17990  catcisolem  18173  prfcl  18265  evlfcl  18284  curf1cl  18290  curfcl  18294  curfuncf  18300  yonedainv  18343  imaid  49960  upciclem3  49974  fucoid  50154  fucorid  50168  precofvallem  50172  termcfuncval  50338