Theorem curfcl 18193

Description: The curry functor of a functor 𝐹:𝐶 × 𝐷⟶𝐸 is a functor curry_F (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curfcl.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfcl.q	⊢ 𝑄 = (𝐷 FuncCat 𝐸)
curfcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfcl.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfcl.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))

Assertion

Ref	Expression
curfcl	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄))

Proof of Theorem curfcl

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

Step	Hyp	Ref	Expression
1		curfcl.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		eqid 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		curfcl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curfcl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curfcl.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		eqid 2729	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		eqid 2729	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8		eqid 2729	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		eqid 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2729	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curfval 18184	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
12		fvex 6871	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
13	12	mptex 7197	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
14	12, 12	mpoex 8058	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) ∈ V
15	13, 14	op1std 7978	. . . . 5 ⊢ (𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ → (1st ‘𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
16	11, 15	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
17	13, 14	op2ndd 7979	. . . . 5 ⊢ (𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ → (2nd ‘𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
18	11, 17	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
19	16, 18	opeq12d 4845	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
20	11, 19	eqtr4d 2767	. 2 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
21		curfcl.q	. . . . 5 ⊢ 𝑄 = (𝐷 FuncCat 𝐸)
22	21	fucbas 17925	. . . 4 ⊢ (𝐷 Func 𝐸) = (Base‘𝑄)
23		eqid 2729	. . . . 5 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
24	21, 23	fuchom 17926	. . . 4 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑄)
25		eqid 2729	. . . 4 ⊢ (Id‘𝑄) = (Id‘𝑄)
26		eqid 2729	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		eqid 2729	. . . 4 ⊢ (comp‘𝑄) = (comp‘𝑄)
28		funcrcl 17825	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
29	5, 28	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
30	29	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
31	21, 4, 30	fuccat 17935	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Cat)
32		opex 5424	. . . . . 6 ⊢ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ ∈ V
33	32	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ ∈ V)
34	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
35	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
36	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
37		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38		eqid 2729	. . . . . 6 ⊢ ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥)
39	1, 2, 34, 35, 36, 6, 37, 38	curf1cl 18189	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
40	33, 16, 39	fmpt2d 7096	. . . 4 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
41		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
42		ovex 7420	. . . . . . 7 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
43	42	mptex 7197	. . . . . 6 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) ∈ V
44	41, 43	fnmpoi 8049	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) Fn ((Base‘𝐶) × (Base‘𝐶))
45	18	fneq1d 6611	. . . . 5 ⊢ (𝜑 → ((2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) Fn ((Base‘𝐶) × (Base‘𝐶))))
46	44, 45	mpbiri 258	. . . 4 ⊢ (𝜑 → (2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
47		fvex 6871	. . . . . . 7 ⊢ (Base‘𝐷) ∈ V
48	47	mptex 7197	. . . . . 6 ⊢ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) ∈ V
49	48	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) ∈ V)
50	18	oveqd 7404	. . . . . 6 ⊢ (𝜑 → (𝑥(2nd ‘𝐺)𝑦) = (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦))
51	41	ovmpt4g 7536	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) ∈ V) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
52	43, 51	mp3an3 1452	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
53	50, 52	sylan9eq 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
54	3	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
55	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
56	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
57		simplrl 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
58		simplrr 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
59		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
60		eqid 2729	. . . . . 6 ⊢ ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) = ((𝑥(2nd ‘𝐺)𝑦)‘𝑔)
61	1, 2, 54, 55, 56, 6, 9, 10, 57, 58, 59, 60, 23	curf2cl 18192	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) ∈ (((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
62	49, 53, 61	fmpt2d 7096	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
63		eqid 2729	. . . . . . . . . 10 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
64	63, 2, 6	xpcbas 18139	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
65		eqid 2729	. . . . . . . . 9 ⊢ (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
66		eqid 2729	. . . . . . . . 9 ⊢ (Id‘𝐸) = (Id‘𝐸)
67		relfunc 17824	. . . . . . . . . . 11 ⊢ Rel ((𝐶 ×c 𝐷) Func 𝐸)
68		1st2ndbr 8021	. . . . . . . . . . 11 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
69	67, 5, 68	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
70	69	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
71		opelxpi 5675	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
72	71	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
73	64, 65, 66, 70, 72	funcid 17832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)))
74	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
75	4	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
76	37	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
77		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
78	63, 74, 75, 2, 6, 8, 10, 65, 76, 77	xpcid 18150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
79	78	fveq2d 6862	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩))
80		df-ov 7390	. . . . . . . . 9 ⊢ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
81	79, 80	eqtr4di 2782	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)))
82	5	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
83	1, 2, 74, 75, 82, 6, 76, 38, 77	curf11 18187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦))
84		df-ov 7390	. . . . . . . . . 10 ⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)
85	83, 84	eqtr2di 2781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘𝐹)‘⟨𝑥, 𝑦⟩) = ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))
86	85	fveq2d 6862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘𝐸)‘((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)))
87	73, 81, 86	3eqtr3d 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)))
88	87	mpteq2dva 5200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))))
89	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
90		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐸) = (Base‘𝐸)
91	90, 66	cidfn 17640	. . . . . . . . 9 ⊢ (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
92	89, 91	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸) Fn (Base‘𝐸))
93		dffn2 6690	. . . . . . . 8 ⊢ ((Id‘𝐸) Fn (Base‘𝐸) ↔ (Id‘𝐸):(Base‘𝐸)⟶V)
94	92, 93	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸):(Base‘𝐸)⟶V)
95		relfunc 17824	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
96		1st2ndbr 8021	. . . . . . . . 9 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
97	95, 39, 96	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
98	6, 90, 97	funcf1 17828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
99		fcompt 7105	. . . . . . 7 ⊢ (((Id‘𝐸):(Base‘𝐸)⟶V ∧ (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸)) → ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))))
100	94, 98, 99	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))))
101	88, 100	eqtr4d 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))) = ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))))
102	2, 9, 8, 34, 37	catidcl 17643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
103		eqid 2729	. . . . . 6 ⊢ ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥))
104	1, 2, 34, 35, 36, 6, 9, 10, 37, 37, 102, 103	curf2 18190	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))))
105	21, 25, 66, 39	fucid 17936	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑄)‘((1st ‘𝐺)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))))
106	101, 104, 105	3eqtr4d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑄)‘((1st ‘𝐺)‘𝑥)))
107	3	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
108	107	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
109	4	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
110	109	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
111	5	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
112	111	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
113		simp21 1207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
114	113	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
115		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑤 ∈ (Base‘𝐷))
116	1, 2, 108, 110, 112, 6, 114, 38, 115	curf11 18187	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤) = (𝑥(1st ‘𝐹)𝑤))
117		df-ov 7390	. . . . . . . . . . 11 ⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)
118	116, 117	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤) = ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩))
119		simp22 1208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
120	119	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
121		eqid 2729	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐺)‘𝑦) = ((1st ‘𝐺)‘𝑦)
122	1, 2, 108, 110, 112, 6, 120, 121, 115	curf11 18187	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤) = (𝑦(1st ‘𝐹)𝑤))
123		df-ov 7390	. . . . . . . . . . 11 ⊢ (𝑦(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩)
124	122, 123	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤) = ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩))
125	118, 124	opeq12d 4845	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩ = ⟨((1st ‘𝐹)‘⟨𝑥, 𝑤⟩), ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩)⟩)
126		simp23 1209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
127	126	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐶))
128		eqid 2729	. . . . . . . . . . 11 ⊢ ((1st ‘𝐺)‘𝑧) = ((1st ‘𝐺)‘𝑧)
129	1, 2, 108, 110, 112, 6, 127, 128, 115	curf11 18187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤) = (𝑧(1st ‘𝐹)𝑤))
130		df-ov 7390	. . . . . . . . . 10 ⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)
131	129, 130	eqtrdi 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤) = ((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))
132	125, 131	oveq12d 7405	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤)) = (⟨((1st ‘𝐹)‘⟨𝑥, 𝑤⟩), ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)))
133		simp3r 1203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
134	133	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
135		eqid 2729	. . . . . . . . . 10 ⊢ ((𝑦(2nd ‘𝐺)𝑧)‘𝑔) = ((𝑦(2nd ‘𝐺)𝑧)‘𝑔)
136	1, 2, 108, 110, 112, 6, 9, 10, 120, 127, 134, 135, 115	curf2val 18191	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤) = (𝑔(⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
137		df-ov 7390	. . . . . . . . 9 ⊢ (𝑔(⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)
138	136, 137	eqtrdi 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤) = ((⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩))
139		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
140	139	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
141		eqid 2729	. . . . . . . . . 10 ⊢ ((𝑥(2nd ‘𝐺)𝑦)‘𝑓) = ((𝑥(2nd ‘𝐺)𝑦)‘𝑓)
142	1, 2, 108, 110, 112, 6, 9, 10, 114, 120, 140, 141, 115	curf2val 18191	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤) = (𝑓(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)((Id‘𝐷)‘𝑤)))
143		df-ov 7390	. . . . . . . . 9 ⊢ (𝑓(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
144	142, 143	eqtrdi 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩))
145	132, 138, 144	oveq123d 7408	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤)) = (((⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st ‘𝐹)‘⟨𝑥, 𝑤⟩), ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)))
146		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
147		eqid 2729	. . . . . . . 8 ⊢ (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
148		eqid 2729	. . . . . . . 8 ⊢ (comp‘𝐸) = (comp‘𝐸)
149	67, 112, 68	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
150		opelxpi 5675	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
151	113, 150	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
152		opelxpi 5675	. . . . . . . . 9 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
153	119, 152	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
154		opelxpi 5675	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
155	126, 154	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
156	6, 7, 10, 110, 115	catidcl 17643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
157	140, 156	opelxpd 5677	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑤(Hom ‘𝐷)𝑤)))
158	63, 2, 6, 9, 7, 114, 115, 120, 115, 146	xpchom2 18147	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑤⟩) = ((𝑥(Hom ‘𝐶)𝑦) × (𝑤(Hom ‘𝐷)𝑤)))
159	157, 158	eleqtrrd 2831	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑤⟩))
160	134, 156	opelxpd 5677	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑔, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑦(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
161	63, 2, 6, 9, 7, 120, 115, 127, 115, 146	xpchom2 18147	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑦, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩) = ((𝑦(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
162	160, 161	eleqtrrd 2831	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑔, ((Id‘𝐷)‘𝑤)⟩ ∈ (⟨𝑦, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩))
163	64, 146, 147, 148, 149, 151, 153, 155, 159, 162	funcco 17833	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = (((⟨𝑦, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st ‘𝐹)‘⟨𝑥, 𝑤⟩), ((1st ‘𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)))
164		eqid 2729	. . . . . . . . . . 11 ⊢ (comp‘𝐷) = (comp‘𝐷)
165	63, 2, 6, 9, 7, 114, 115, 120, 115, 26, 164, 147, 127, 115, 140, 156, 134, 156	xpcco2 18148	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩) = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤))⟩)
166	6, 7, 10, 110, 115, 164, 115, 156	catlid 17644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤)) = ((Id‘𝐷)‘𝑤))
167	166	opeq2d 4844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤))⟩ = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
168	165, 167	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩) = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
169	168	fveq2d 6862	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩))
170		df-ov 7390	. . . . . . . 8 ⊢ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
171	169, 170	eqtr4di 2782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
172	145, 163, 171	3eqtr2rd 2771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤)))
173	172	mpteq2dva 5200	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑤 ∈ (Base‘𝐷) ↦ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤))) = (𝑤 ∈ (Base‘𝐷) ↦ ((((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤))))
174	2, 9, 26, 107, 113, 119, 126, 139, 133	catcocl 17646	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
175		eqid 2729	. . . . . 6 ⊢ ((𝑥(2nd ‘𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd ‘𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
176	1, 2, 107, 109, 111, 6, 9, 10, 113, 126, 174, 175	curf2 18190	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd ‘𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (𝑤 ∈ (Base‘𝐷) ↦ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤))))
177	1, 2, 107, 109, 111, 6, 9, 10, 113, 119, 139, 141, 23	curf2cl 18192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑓) ∈ (((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
178	1, 2, 107, 109, 111, 6, 9, 10, 119, 126, 133, 135, 23	curf2cl 18192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd ‘𝐺)𝑧)‘𝑔) ∈ (((1st ‘𝐺)‘𝑦)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑧)))
179	21, 23, 6, 148, 27, 177, 178	fucco 17927	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd ‘𝐺)𝑧)‘𝑔)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝑄)((1st ‘𝐺)‘𝑧))((𝑥(2nd ‘𝐺)𝑦)‘𝑓)) = (𝑤 ∈ (Base‘𝐷) ↦ ((((𝑦(2nd ‘𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤), ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))(((𝑥(2nd ‘𝐺)𝑦)‘𝑓)‘𝑤))))
180	173, 176, 179	3eqtr4d 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd ‘𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘𝐺)𝑧)‘𝑔)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐺)‘𝑦)⟩(comp‘𝑄)((1st ‘𝐺)‘𝑧))((𝑥(2nd ‘𝐺)𝑦)‘𝑓)))
181	2, 22, 9, 24, 8, 25, 26, 27, 3, 31, 40, 46, 62, 106, 180	isfuncd 17827	. . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝑄)(2nd ‘𝐺))
182		df-br 5108	. . 3 ⊢ ((1st ‘𝐺)(𝐶 Func 𝑄)(2nd ‘𝐺) ↔ ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∈ (𝐶 Func 𝑄))
183	181, 182	sylib 218	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∈ (𝐶 Func 𝑄))
184	20, 183	eqeltrd 2828	1 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636   ∘ ccom 5642  Rel wrel 5643   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Idccid 17626   Func cfunc 17816   Nat cnat 17906   FuncCat cfuc 17907   ×_c cxpc 18129   curry_F ccurf 18171

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-rmo 3354  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-cid 17630  df-func 17820  df-nat 17908  df-fuc 17909  df-xpc 18133  df-curf 18175

This theorem is referenced by:  uncfcurf  18200  diagcl  18202  curf2ndf  18208  yoncl  18223  tposcurfcl  49292