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Theorem fucidcl 17935

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 17830	. . . . . . . 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 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		fucidcl.x	. . . . . . 7 ⊢ 1 = (Id‘𝐷)
7	5, 6	cidfn 17645	. . . . . 6 ⊢ (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 1 Fn (Base‘𝐷))
9		dffn2 6670	. . . . 5 ⊢ ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
10	8, 9	sylib 218	. . . 4 ⊢ (𝜑 → 1 :(Base‘𝐷)⟶V)
11		eqid 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		relfunc 17829	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
13		1st2ndbr 7995	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	12, 1, 13	sylancr 588	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
15	11, 5, 14	funcf1 17833	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16		fcompt 7086	. . . 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 585	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1^st ‘𝐹)‘𝑥))))
18		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
20	15	ffvelcdmda 7036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	5, 18, 6, 19, 20	catidcl 17648	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
22	21	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
23		fvex 6853	. . . . 5 ⊢ (Base‘𝐶) ∈ V
24		mptelixpg 8883	. . . . 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 2836	. 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 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
30	29, 20	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
31		eqid 2736	. . . . . 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 1197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
34	32, 33	ffvelcdmd 7037	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
35		eqid 2736	. . . . . . . 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 17835	. . . . . . 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 1198	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
39	37, 38	ffvelcdmd 7037	. . . . . 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 17649	. . . . 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 17650	. . . . 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 2774	. . . 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 6939	. . . . . 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 585	. . . . 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 7382	. . . 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 6939	. . . . . 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 585	. . . . 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 7383	. . . 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 2781	. . 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 3183	. 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 17918	. 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 714	1 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 ⟨cop 4573 class class class wbr 5085 ↦ cmpt 5166 ∘ ccom 5635 Rel wrel 5636 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 Xcixp 8845 Basecbs 17179 Hom chom 17231 compcco 17232 Catccat 17630 Idccid 17631 Func cfunc 17821 Nat cnat 17911 FuncCat cfuc 17912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-map 8775 df-ixp 8846 df-cat 17634 df-cid 17635 df-func 17825 df-nat 17913

This theorem is referenced by: fuclid 17936 fucrid 17937 fuccatid 17939 fucolid 49836 fucorid 49837 precofvalALT 49843 fucoppcid 49883

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