Theorem fuccocl 18135

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 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2761	. . . 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 18133	. . 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 2761	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	2	natrcl 18121	. . . . . . . . . . 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 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		funcrcl 18031	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
16	15	simprd 501	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		relfunc 18030	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8051	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
20	18, 13, 19	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
21	3, 9, 20	funcf1 18034	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23	2	natrcl 18121	. . . . . . . . . . 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 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
26		1st2ndbr 8051	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
27	18, 25, 26	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
28	3, 9, 27	funcf1 18034	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
29	28	ffvelcdmda 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30	24	simprd 501	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
31		1st2ndbr 8051	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
32	18, 30, 31	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
33	3, 9, 32	funcf1 18034	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	2, 6	nat1st2nd 18122	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
36	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
37		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	2, 36, 3, 10, 37	natcl 18124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑥)))
39	2, 7	nat1st2nd 18122	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
40	39	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
41	2, 40, 3, 10, 37	natcl 18124	. . . . . 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 17852	. . . . 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 3155	. . . 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 8956	. . . . 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 2861	. 2 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
49	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
50	21	adantr 486	. . . . . . 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 7083	. . . . . 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 7083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
55	28	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
56	55, 53	ffvelcdmd 7083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
57		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
58	20	adantr 486	. . . . . . . 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 18036	. . . . . . 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 7083	. . . . . 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 486	. . . . . . 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 18124	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
65	64, 53	ffvelcdmd 7083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐻)‘𝑦) ∈ (Base‘𝐷))
66	39	adantr 486	. . . . . . 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 18124	. . . . . 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 17853	. . . . 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 18126	. . . . . . 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 7434	. . . . . 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 7083	. . . . . . 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 18124	. . . . . . 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 486	. . . . . . . . 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 18036	. . . . . . . 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 7083	. . . . . . 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 17853	. . . . . 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 18126	. . . . . . 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 7433	. . . . . 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 2802	. . . . 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 7083	. . . . . 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 18124	. . . . . 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 486	. . . . . . . 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 18036	. . . . . . 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 7083	. . . . . 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 17853	. . . . 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 2800	. . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (𝐹𝑁𝐺))
88	7	adantr 486	. . . . . 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 18134	. . . . 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 7433	. . . 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 18134	. . . . 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 7434	. . . 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 2806	. . 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 3209	. 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 18119	. 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 726	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186  Rel wrel 5656  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  Xcixp 8918  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831   Func cfunc 18022   Nat cnat 18112   FuncCat cfuc 18113

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  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 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-cat 17835  df-func 18026  df-nat 18114  df-fuc 18115

This theorem is used by:  fucass  18139  fuccatid  18140  evlfcllem  18388  yonedalem3b  18446  xpcfuccocl  50334  fucoppcco  50486