fucidcl - Metamath Proof Explorer

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Theorem fucidcl 17981

Description: The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fucidcl.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucidcl.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucidcl.x	⊢ 1 = (Id‘𝐷)
fucidcl.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
fucidcl	⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))

Proof of Theorem fucidcl Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucidcl.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		funcrcl 17876	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		eqid 2735	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		fucidcl.x	. . . . . . 7 ⊢ 1 = (Id‘𝐷)
7	5, 6	cidfn 17691	. . . . . 6 ⊢ (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 1 Fn (Base‘𝐷))
9		dffn2 6708	. . . . 5 ⊢ ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
10	8, 9	sylib 218	. . . 4 ⊢ (𝜑 → 1 :(Base‘𝐷)⟶V)
11		eqid 2735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		relfunc 17875	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
13		1st2ndbr 8041	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	12, 1, 13	sylancr 587	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
15	11, 5, 14	funcf1 17879	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16		fcompt 7123	. . . 4 ⊢ (( 1 :(Base‘𝐷)⟶V ∧ (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ( 1 ∘ (1^st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))))
17	10, 15, 16	syl2anc 584	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))))
18		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
20	15	ffvelcdmda 7074	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	5, 18, 6, 19, 20	catidcl 17694	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
22	21	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
23		fvex 6889	. . . . 5 ⊢ (Base‘𝐶) ∈ V
24		mptelixpg 8949	. . . . 5 ⊢ ((Base‘𝐶) ∈ V → ((𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))))
25	23, 24	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
26	22, 25	sylibr 234	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
27	17, 26	eqeltrd 2834	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
28	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29		simpr1 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
30	29, 20	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
31		eqid 2735	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		simpr2 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
34	32, 33	ffvelcdmd 7075	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
35		eqid 2735	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
36	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
37	11, 35, 18, 36, 29, 33	funcf2 17881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2^nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
38		simpr3 1197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
39	37, 38	ffvelcdmd 7075	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2^nd ‘𝐹)𝑦)‘𝑓) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
40	5, 18, 6, 28, 30, 31, 34, 39	catlid 17695	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ‘((1^st ‘𝐹)‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = ((𝑥(2^nd ‘𝐹)𝑦)‘𝑓))
41	5, 18, 6, 28, 30, 31, 34, 39	catrid 17696	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))( 1 ‘((1^st ‘𝐹)‘𝑥))) = ((𝑥(2^nd ‘𝐹)𝑦)‘𝑓))
42	40, 41	eqtr4d 2773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ‘((1^st ‘𝐹)‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))( 1 ‘((1^st ‘𝐹)‘𝑥))))
43		fvco3 6978	. . . . . 6 ⊢ (((1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐹))‘𝑦) = ( 1 ‘((1^st ‘𝐹)‘𝑦)))
44	32, 33, 43	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1^st ‘𝐹))‘𝑦) = ( 1 ‘((1^st ‘𝐹)‘𝑦)))
45	44	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((( 1 ∘ (1^st ‘𝐹))‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (( 1 ‘((1^st ‘𝐹)‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)))
46		fvco3 6978	. . . . . 6 ⊢ (((1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐹))‘𝑥) = ( 1 ‘((1^st ‘𝐹)‘𝑥)))
47	32, 29, 46	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1^st ‘𝐹))‘𝑥) = ( 1 ‘((1^st ‘𝐹)‘𝑥)))
48	47	oveq2d 7421	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))(( 1 ∘ (1^st ‘𝐹))‘𝑥)) = (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))( 1 ‘((1^st ‘𝐹)‘𝑥))))
49	42, 45, 48	3eqtr4d 2780	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((( 1 ∘ (1^st ‘𝐹))‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))(( 1 ∘ (1^st ‘𝐹))‘𝑥)))
50	49	ralrimivvva 3190	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)((( 1 ∘ (1^st ‘𝐹))‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))(( 1 ∘ (1^st ‘𝐹))‘𝑥)))
51		fucidcl.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
52	51, 11, 35, 18, 31, 1, 1	isnat2 17964	. 2 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹) ↔ (( 1 ∘ (1^st ‘𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)((( 1 ∘ (1^st ‘𝐹))‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑦))(( 1 ∘ (1^st ‘𝐹))‘𝑥)))))
53	27, 50, 52	mpbir2and 713	1 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 ⟨cop 4607 class class class wbr 5119 ↦ cmpt 5201 ∘ ccom 5658 Rel wrel 5659 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 1^st c1st 7986 2^nd c2nd 7987 Xcixp 8911 Basecbs 17228 Hom chom 17282 compcco 17283 Catccat 17676 Idccid 17677 Func cfunc 17867 Nat cnat 17957 FuncCat cfuc 17958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-map 8842 df-ixp 8912 df-cat 17680 df-cid 17681 df-func 17871 df-nat 17959

This theorem is referenced by: fuclid 17982 fucrid 17983 fuccatid 17985 fucolid 49272 fucorid 49273 precofvalALT 49279

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