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Theorem fucidcl 17229

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 17127	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simprd 498	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		eqid 2821	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		fucidcl.x	. . . . . . 7 ⊢ 1 = (Id‘𝐷)
7	5, 6	cidfn 16944	. . . . . 6 ⊢ (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 1 Fn (Base‘𝐷))
9		dffn2 6510	. . . . 5 ⊢ ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
10	8, 9	sylib 220	. . . 4 ⊢ (𝜑 → 1 :(Base‘𝐷)⟶V)
11		eqid 2821	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		relfunc 17126	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
13		1st2ndbr 7735	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	12, 1, 13	sylancr 589	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
15	11, 5, 14	funcf1 17130	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16		fcompt 6889	. . . 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 586	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))))
18		eqid 2821	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
20	15	ffvelrnda 6845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	5, 18, 6, 19, 20	catidcl 16947	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
22	21	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
23		fvex 6677	. . . . 5 ⊢ (Base‘𝐶) ∈ V
24		mptelixpg 8493	. . . . 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 236	. . 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 2913	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
28	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29		simpr1 1190	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
30	29, 20	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
31		eqid 2821	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	15	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		simpr2 1191	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
34	32, 33	ffvelrnd 6846	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
35		eqid 2821	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
36	14	adantr 483	. . . . . . . 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 17132	. . . . . . 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 1192	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
39	37, 38	ffvelrnd 6846	. . . . . 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 16948	. . . . 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 16949	. . . . 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 2859	. . . 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 6754	. . . . . 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 586	. . . . 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 7165	. . . 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 6754	. . . . . 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 586	. . . . 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 7166	. . . 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 2866	. . 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 3192	. 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 17212	. 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 711	1 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⟨cop 4566 class class class wbr 5058 ↦ cmpt 5138 ∘ ccom 5553 Rel wrel 5554 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 1^st c1st 7681 2^nd c2nd 7682 Xcixp 8455 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Idccid 16930 Func cfunc 17118 Nat cnat 17205 FuncCat cfuc 17206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-ixp 8456 df-cat 16933 df-cid 16934 df-func 17122 df-nat 17207

This theorem is referenced by: fuclid 17230 fucrid 17231 fuccatid 17233

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