Theorem fuccocl 18013

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 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2736	. . . 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 18011	. . 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 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	2	natrcl 17999	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
12	6, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
13	12	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
14		funcrcl 17909	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
16	15	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		relfunc 17908	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8068	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
20	18, 13, 19	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
21	3, 9, 20	funcf1 17912	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23	2	natrcl 17999	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
24	7, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
25	24	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
26		1st2ndbr 8068	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
27	18, 25, 26	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
28	3, 9, 27	funcf1 17912	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
29	28	ffvelcdmda 7103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30	24	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
31		1st2ndbr 8068	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
32	18, 30, 31	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐻)(𝐶 Func 𝐷)(2nd ‘𝐻))
33	3, 9, 32	funcf1 17912	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	2, 6	nat1st2nd 18000	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
36	35	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
37		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	2, 36, 3, 10, 37	natcl 18002	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐺)‘𝑥)))
39	2, 7	nat1st2nd 18000	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
40	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩𝑁⟨(1st ‘𝐻), (2nd ‘𝐻)⟩))
41	2, 40, 3, 10, 37	natcl 18002	. . . . . 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 17729	. . . . 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 3145	. . . 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 6918	. . . . 5 ⊢ (Base‘𝐶) ∈ V
45		mptelixpg 8976	. . . . 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 234	. . 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 2840	. 2 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐻)‘𝑥)))
49	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
50	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr1 1194	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
53		simpr2 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
54	50, 53	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
55	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
56	55, 53	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
57		eqid 2736	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
58	20	adantr 480	. . . . . . . 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 17914	. . . . . . 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 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
61	59, 60	ffvelcdmd 7104	. . . . . 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 480	. . . . . . 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 18002	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
65	64, 53	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st ‘𝐻)‘𝑦) ∈ (Base‘𝐷))
66	39	adantr 480	. . . . . . 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 18002	. . . . . 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 17730	. . . . 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 18004	. . . . . . 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 7448	. . . . . 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 7104	. . . . . . 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 18002	. . . . . . 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 480	. . . . . . . . 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 17914	. . . . . . . 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 7104	. . . . . . 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 17730	. . . . . 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 18004	. . . . . . 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 7447	. . . . . 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 2782	. . . . 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 7104	. . . . . 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 18002	. . . . . 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 480	. . . . . . . 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 17914	. . . . . . 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 7104	. . . . . 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 17730	. . . . 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 2780	. . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (𝐹𝑁𝐺))
88	7	adantr 480	. . . . . 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 18012	. . . . 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 7447	. . . 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 18012	. . . . 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 7448	. . . 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 2786	. . 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 3204	. 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 17997	. 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 713	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479  ⟨cop 4631   class class class wbr 5142   ↦ cmpt 5224  Rel wrel 5689  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  Xcixp 8938  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708   Func cfunc 17900   Nat cnat 17990   FuncCat cfuc 17991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-func 17904  df-nat 17992  df-fuc 17993

This theorem is referenced by:  fucass  18017  fuccatid  18018  evlfcllem  18267  yonedalem3b  18325  xpcfuccocl  48981