Theorem funcid 17783

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 7370	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋))
3		fveq2 6828	. . . 4 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
4	2, 3	fveq12d 6835	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = ((𝑋𝐺𝑋)‘( 1 ‘𝑋)))
5		2fveq3 6833	. . 3 ⊢ (𝑥 = 𝑋 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑋)))
6	4, 5	eqeq12d 2747	. 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 2731	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
10		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12		funcid.1	. . . . . 6 ⊢ 1 = (Id‘𝐷)
13		funcid.i	. . . . . 6 ⊢ 𝐼 = (Id‘𝐸)
14		eqid 2731	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
15		eqid 2731	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
16		df-br 5094	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
17	7, 16	sylib 218	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
18		funcrcl 17776	. . . . . . . 8 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
20	19	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	19	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
22	8, 9, 10, 11, 12, 13, 14, 15, 20, 21	isfunc 17777	. . . . 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 232	. . . 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 1144	. . 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 482	. . . 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 3069	. . 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 3573	1 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⟨cop 4581   class class class wbr 5093   × cxp 5617  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926   ↑_m cmap 8756  Xcixp 8827  Basecbs 17126  Hom chom 17178  compcco 17179  Catccat 17576  Idccid 17577   Func cfunc 17767

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-ixp 8828  df-func 17771

This theorem is referenced by:  funcsect  17785  funcoppc  17788  cofucl  17801  funcres  17809  fthsect  17840  catcisolem  18023  prfcl  18115  evlfcl  18134  curf1cl  18140  curfcl  18144  curfuncf  18150  yonedainv  18193  imaid  49260  upciclem3  49274  fucoid  49454  fucorid  49468  precofvallem  49472  termcfuncval  49638