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Theorem fuccocl 17929

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 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2729	. . . 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 17927	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
9		eqid 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	2	natrcl 17915	. . . . . . . . . . 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 17825	. . . . . . . . 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 17824	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8021	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
20	18, 13, 19	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
21	3, 9, 20	funcf1 17828	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23	2	natrcl 17915	. . . . . . . . . . 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 8021	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
27	18, 25, 26	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
28	3, 9, 27	funcf1 17828	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
29	28	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30	24	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
31		1st2ndbr 8021	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
32	18, 30, 31	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	3, 9, 32	funcf1 17828	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	2, 6	nat1st2nd 17916	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
36	35	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
37		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	2, 36, 3, 10, 37	natcl 17918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
39	2, 7	nat1st2nd 17916	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
40	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
41	2, 40, 3, 10, 37	natcl 17918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
42	9, 10, 4, 17, 22, 29, 34, 38, 41	catcocl 17646	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
43	42	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
44		fvex 6871	. . . . 5 ⊢ (Base‘𝐶) ∈ V
45		mptelixpg 8908	. . . . 5 ⊢ ((Base‘𝐶) ∈ V → ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥))))
46	44, 45	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
47	43, 46	sylibr 234	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
48	8, 47	eqeltrd 2828	. 2 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘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))) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr1 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
53		simpr2 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
54	50, 53	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
55	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
56	55, 53	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
57		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
58	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
59	3, 57, 10, 58, 51, 53	funcf2 17830	. . . . . . 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)))
60		simpr3 1197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
61	59, 60	ffvelcdmd 7057	. . . . . 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)))
62	35	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
63	2, 62, 3, 10, 53	natcl 17918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
64	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
65	64, 53	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐻)‘𝑦) ∈ (Base‘𝐷))
66	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
67	2, 66, 3, 10, 53	natcl 17918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆‘𝑦) ∈ (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑦)))
68	9, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67	catass 17647	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓))))
69	2, 62, 3, 57, 4, 51, 53, 60	nati 17920	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑅‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐺)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))(𝑅‘𝑥)))
70	69	oveq2d 7403	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓))) = ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(((𝑥(2^nd ‘𝐺)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))(𝑅‘𝑥))))
71	55, 51	ffvelcdmd 7057	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
72	2, 62, 3, 10, 51	natcl 17918	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
73	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
74	3, 57, 10, 73, 51, 53	funcf2 17830	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2^nd ‘𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
75	74, 60	ffvelcdmd 7057	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2^nd ‘𝐺)𝑦)‘𝑓) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
76	9, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67	catass 17647	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐺)𝑦)‘𝑓))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(((𝑥(2^nd ‘𝐺)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))(𝑅‘𝑥))))
77	2, 66, 3, 57, 4, 51, 53, 60	nati 17920	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐺)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑆‘𝑥)))
78	77	oveq1d 7402	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐺)𝑦)‘𝑓))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = ((((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑥)))
79	70, 76, 78	3eqtr2d 2770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑅‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓))) = ((((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑥)))
80	64, 51	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
81	2, 66, 3, 10, 51	natcl 17918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
82	32	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
83	3, 57, 10, 82, 51, 53	funcf2 17830	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2^nd ‘𝐻)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1^st ‘𝐻)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑦)))
84	83, 60	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2^nd ‘𝐻)𝑦)‘𝑓) ∈ (((1^st ‘𝐻)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑦)))
85	9, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84	catass 17647	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑆‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑥)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
86	68, 79, 85	3eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘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 17928	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦) = ((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑦)))
90	89	oveq1d 7402	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑆‘𝑦)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))(𝑅‘𝑦))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)))
91	1, 2, 3, 4, 5, 87, 88, 51	fuccoval 17928	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥) = ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
92	91	oveq2d 7403	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
93	86, 90, 92	3eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))
94	93	ralrimivvva 3183	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))
95	2, 3, 57, 10, 4, 13, 30	isnat2 17913	. 2 ⊢ (𝜑 → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻) ↔ ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘𝑓)) = (((𝑥(2^nd ‘𝐻)𝑦)‘𝑓)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))))
96	48, 94, 95	mpbir2and 713	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 Rel wrel 5643 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 Xcixp 8870 Basecbs 17179 Hom chom 17231 compcco 17232 Catccat 17625 Func cfunc 17816 Nat cnat 17906 FuncCat cfuc 17907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-func 17820 df-nat 17908 df-fuc 17909

This theorem is referenced by: fucass 17933 fuccatid 17934 evlfcllem 18182 yonedalem3b 18240 xpcfuccocl 49243 fucoppcco 49395

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