Theorem funcid 17142

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 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
2	1, 1	oveq12d 7176	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3		fveq2 6672	. . . 4 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
4	2, 3	fveq12d 6679	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = ((𝑋𝐺𝑋)‘( 1 ‘𝑋)))
5		2fveq3 6677	. . 3 ⊢ (𝑥 = 𝑋 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑋)))
6	4, 5	eqeq12d 2839	. 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 2823	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
10		eqid 2823	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11		eqid 2823	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12		funcid.1	. . . . . 6 ⊢ 1 = (Id‘𝐷)
13		funcid.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐸)
14		eqid 2823	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
15		eqid 2823	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
16		df-br 5069	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
17	7, 16	sylib 220	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18		funcrcl 17135	. . . . . . . 8 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
20	19	simpld 497	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	19	simprd 498	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
22	8, 9, 10, 11, 12, 13, 14, 15, 20, 21	isfunc 17136	. . . . 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 234	. . . 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 1140	. . 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 485	. . . 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 3162	. . 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 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
28		funcid.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
29	6, 27, 28	rspcdva 3627	1 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟨cop 4575   class class class wbr 5068   × cxp 5555  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690   ↑_m cmap 8408  Xcixp 8463  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938   Func cfunc 17126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-ixp 8464  df-func 17130

This theorem is referenced by:  funcsect  17144  funcoppc  17147  cofucl  17160  funcres  17168  fthsect  17197  catcisolem  17368  prfcl  17455  evlfcl  17474  curf1cl  17480  curfcl  17484  curfuncf  17490  yonedainv  17533