Theorem fuccocl 18025

Description: The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fuccocl.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuccocl.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fuccocl.x	⊢ ∙ = (comp‘𝑄)
fuccocl.r	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
fuccocl.s	⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))

Assertion

Ref	Expression
fuccocl	⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Proof of Theorem fuccocl

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuccocl.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		fuccocl.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
3		eqid 2763	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2763	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		fuccocl.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
6		fuccocl.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
7		fuccocl.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
8	1, 2, 3, 4, 5, 6, 7	fucco 18023	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
9		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	2	natrcl 18011	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
12	6, 11	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
13	12	simpld 499	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		funcrcl 17921	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
16	15	simprd 500	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		relfunc 17920	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8040	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
20	18, 13, 19	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
21	3, 9, 20	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23	2	natrcl 18011	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
24	7, 23	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
25	24	simpld 499	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
26		1st2ndbr 8040	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
27	18, 25, 26	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
28	3, 9, 27	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
29	28	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30	24	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
31		1st2ndbr 8040	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
32	18, 30, 31	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
33	3, 9, 32	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	2, 6	nat1st2nd 18012	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
36	35	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
37		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	2, 36, 3, 10, 37	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑥)))
39	2, 7	nat1st2nd 18012	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
40	39	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
41	2, 40, 3, 10, 37	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
42	9, 10, 4, 17, 22, 29, 34, 38, 41	catcocl 17742	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
43	42	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
44		fvex 6896	. . . . 5 ⊢ (Base‘𝐶) ∈ V
45		mptelixpg 8934	. . . . 5 ⊢ ((Base‘𝐶) ∈ V → ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥))))
46	44, 45	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
47	43, 46	sylibr 237	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
48	8, 47	eqeltrd 2863	. 2 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
49	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
50	21	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr1 1213	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
53		simpr2 1214	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
54	50, 53	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
55	28	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
56	55, 53	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
57		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
58	20	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
59	3, 57, 10, 58, 51, 53	funcf2 17926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
60		simpr3 1215	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
61	59, 60	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
62	35	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
63	2, 62, 3, 10, 53	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅‘𝑦) ∈ (((1st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1st ‘𝐺)‘𝑦)))
64	33	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
65	64, 53	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐻)‘𝑦) ∈ (Base‘𝐷))
66	39	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
67	2, 66, 3, 10, 53	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆‘𝑦) ∈ (((1st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1st ‘𝐻)‘𝑦)))
68	9, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67	catass 17743	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑦), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))))
69	2, 62, 3, 57, 4, 51, 53, 60	nati 18016	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑅‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐺)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))(𝑅‘𝑥)))
70	69	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) = ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))(𝑅‘𝑥))))
71	55, 51	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
72	2, 62, 3, 10, 51	natcl 18014	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑥)))
73	27	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
74	3, 57, 10, 73, 51, 53	funcf2 17926	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd ‘𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑦)))
75	74, 60	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑓) ∈ (((1st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑦)))
76	9, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67	catass 17743	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐺)𝑦)‘𝑓))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))(𝑅‘𝑥))))
77	2, 66, 3, 57, 4, 51, 53, 60	nati 18016	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐺)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑆‘𝑥)))
78	77	oveq1d 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐺)𝑦)‘𝑓))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = ((((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑥)))
79	70, 76, 78	3eqtr2d 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) = ((((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑥)))
80	64, 51	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
81	2, 66, 3, 10, 51	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆‘𝑥) ∈ (((1st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
82	32	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
83	3, 57, 10, 82, 51, 53	funcf2 17926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd ‘𝐻)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐻)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑦)))
84	83, 60	ffvelcdmd 7082	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd ‘𝐻)𝑦)‘𝑓) ∈ (((1st ‘𝐻)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑦)))
85	9, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84	catass 17743	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
86	68, 79, 85	3eqtrd 2802	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑦), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
87	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (𝐹𝑁𝐺))
88	7	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (𝐺𝑁𝐻))
89	1, 2, 3, 4, 5, 87, 88, 53	fuccoval 18024	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦) = ((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑦), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑦)))
90	89	oveq1d 7427	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑆‘𝑦)(⟨((1st ‘𝐹)‘𝑦), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
91	1, 2, 3, 4, 5, 87, 88, 51	fuccoval 18024	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥) = ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
92	91	oveq2d 7428	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
93	86, 90, 92	3eqtr4d 2808	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))
94	93	ralrimivvva 3211	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))
95	2, 3, 57, 10, 4, 13, 30	isnat2 18009	. 2 ⊢ (𝜑 → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻) ↔ ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2nd ‘𝐻)𝑦)‘𝑓)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))))
96	48, 94, 95	mpbir2and 725	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  ⟨cop 4596   class class class wbr 5110   ↦ cmpt 5193  Rel wrel 5668  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Xcixp 8896  Basecbs 17270  Hom chom 17322  compcco 17323  Catccat 17721   Func cfunc 17912   Nat cnat 18002   FuncCat cfuc 18003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-slot 17243  df-ndx 17255  df-base 17271  df-hom 17335  df-cco 17336  df-cat 17725  df-func 17916  df-nat 18004  df-fuc 18005

This theorem is referenced by:  fucass  18029  fuccatid  18030  evlfcllem  18278  yonedalem3b  18336  xpcfuccocl  50018  fucoppcco  50170