fucidcl - Metamath Proof Explorer

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Theorem fucidcl 18063

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 17958	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		eqid 2762	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		fucidcl.x	. . . . . . 7 ⊢ 1 = (Id‘𝐷)
7	5, 6	cidfn 17773	. . . . . 6 ⊢ (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
8	4, 7	syl 18	. . . . 5 ⊢ (𝜑 → 1 Fn (Base‘𝐷))
9		dffn2 6708	. . . . 5 ⊢ ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
10	8, 9	sylib 221	. . . 4 ⊢ (𝜑 → 1 :(Base‘𝐷)⟶V)
11		eqid 2762	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		relfunc 17957	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
13		1st2ndbr 8043	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	12, 1, 13	sylancr 599	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
15	11, 5, 14	funcf1 17961	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16		fcompt 7131	. . . 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 596	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))))
18		eqid 2762	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
20	15	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	5, 18, 6, 19, 20	catidcl 17776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
22	21	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
23		fvex 6895	. . . . 5 ⊢ (Base‘𝐶) ∈ V
24		mptelixpg 8946	. . . . 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 237	. . 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 2862	. 2 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
28	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29		simpr1 1213	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
30	29, 20	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
31		eqid 2762	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33		simpr2 1214	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
34	32, 33	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
35		eqid 2762	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
36	14	adantr 486	. . . . . . . 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 17963	. . . . . . 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 1215	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
39	37, 38	ffvelcdmd 7082	. . . . . 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 17777	. . . . 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 17778	. . . . 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 2800	. . . 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 6982	. . . . . 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 596	. . . . 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 7432	. . . 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 6982	. . . . . 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 596	. . . . 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 7433	. . . 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 2807	. . 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 3210	. 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 18046	. 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 726	1 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 Vcvv 3453 ⟨cop 4593 class class class wbr 5107 ↦ cmpt 5190 ∘ ccom 5663 Rel wrel 5664 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 1^st c1st 7988 2^nd c2nd 7989 Xcixp 8908 Basecbs 17307 Hom chom 17359 compcco 17360 Catccat 17758 Idccid 17759 Func cfunc 17949 Nat cnat 18039 FuncCat cfuc 18040

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-map 8832 df-ixp 8909 df-cat 17762 df-cid 17763 df-func 17953 df-nat 18041

This theorem is used by: fuclid 18064 fucrid 18065 fuccatid 18067 fucolid 50295 fucorid 50296 precofvalALT 50302 fucoppcid 50342

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