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Theorem curf2cl 17949

Description: The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)

**Hypotheses**
Ref	Expression
curf2.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curry_F 𝐹)
curf2.a	⊢ 𝐴 = (Base‘𝐶)
curf2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curf2.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curf2.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸))
curf2.b	⊢ 𝐵 = (Base‘𝐷)
curf2.h	⊢ 𝐻 = (Hom ‘𝐶)
curf2.i	⊢ 𝐼 = (Id‘𝐷)
curf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
curf2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
curf2.l	⊢ 𝐿 = ((𝑋(2^nd ‘𝐺)𝑌)‘𝐾)
curf2.n	⊢ 𝑁 = (𝐷 Nat 𝐸)

**Assertion**
Ref	Expression
curf2cl	⊢ (𝜑 → 𝐿 ∈ (((1^st ‘𝐺)‘𝑋)𝑁((1^st ‘𝐺)‘𝑌)))

Proof of Theorem curf2cl Dummy variables 𝑧 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf2.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curry_F 𝐹)
2		curf2.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
3		curf2.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curf2.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curf2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸))
6		curf2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
7		curf2.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
8		curf2.i	. . . 4 ⊢ 𝐼 = (Id‘𝐷)
9		curf2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
10		curf2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
11		curf2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
12		curf2.l	. . . 4 ⊢ 𝐿 = ((𝑋(2^nd ‘𝐺)𝑌)‘𝐾)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	curf2 17947	. . 3 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
14		eqid 2738	. . . . . . . . . 10 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
15	14, 2, 6	xpcbas 17895	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
16		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
17		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
18		relfunc 17577	. . . . . . . . . . 11 ⊢ Rel ((𝐶 ×_c 𝐷) Func 𝐸)
19		1st2ndbr 7883	. . . . . . . . . . 11 ⊢ ((Rel ((𝐶 ×_c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸)) → (1^st ‘𝐹)((𝐶 ×_c 𝐷) Func 𝐸)(2^nd ‘𝐹))
20	18, 5, 19	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐹)((𝐶 ×_c 𝐷) Func 𝐸)(2^nd ‘𝐹))
21	20	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (1^st ‘𝐹)((𝐶 ×_c 𝐷) Func 𝐸)(2^nd ‘𝐹))
22		opelxpi 5626	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
23	9, 22	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24		opelxpi 5626	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
25	10, 24	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
26	15, 16, 17, 21, 23, 25	funcf2 17583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩):(⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩)⟶(((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)))
27		eqid 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
28	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐴)
29		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
30	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑌 ∈ 𝐴)
31	14, 2, 6, 7, 27, 28, 29, 30, 29, 16	xpchom2 17903	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
32	31	feq2d 6586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩):(⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩)⟶(((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)) ↔ (⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩))))
33	26, 32	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)))
34	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ (𝑋𝐻𝑌))
35	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐷 ∈ Cat)
36	6, 27, 8, 35, 29	catidcl 17391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
37	33, 34, 36	fovrnd 7444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) ∈ (((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)))
38	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐶 ∈ Cat)
39	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸))
40		eqid 2738	. . . . . . . . 9 ⊢ ((1^st ‘𝐺)‘𝑋) = ((1^st ‘𝐺)‘𝑋)
41	1, 2, 38, 35, 39, 6, 28, 40, 29	curf11 17944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1^st ‘𝐹)𝑧))
42		df-ov 7278	. . . . . . . 8 ⊢ (𝑋(1^st ‘𝐹)𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)
43	41, 42	eqtrdi 2794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩))
44		eqid 2738	. . . . . . . . 9 ⊢ ((1^st ‘𝐺)‘𝑌) = ((1^st ‘𝐺)‘𝑌)
45	1, 2, 38, 35, 39, 6, 30, 44, 29	curf11 17944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1^st ‘𝐹)𝑧))
46		df-ov 7278	. . . . . . . 8 ⊢ (𝑌(1^st ‘𝐹)𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)
47	45, 46	eqtrdi 2794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩))
48	43, 47	oveq12d 7293	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)) = (((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)))
49	37, 48	eleqtrrd 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) ∈ (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
50	49	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) ∈ (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
51	6	fvexi 6788	. . . . 5 ⊢ 𝐵 ∈ V
52		mptelixpg 8723	. . . . 5 ⊢ (𝐵 ∈ V → ((𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) ∈ (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧))))
53	51, 52	ax-mp 5	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) ∈ (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
54	50, 53	sylibr 233	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
55	13, 54	eqeltrd 2839	. 2 ⊢ (𝜑 → 𝐿 ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
56		eqid 2738	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
57	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
58	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋 ∈ 𝐴)
59		eqid 2738	. . . . . . . . . 10 ⊢ (comp‘𝐶) = (comp‘𝐶)
60	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌 ∈ 𝐴)
61	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌))
62	2, 7, 56, 57, 58, 59, 60, 61	catrid 17393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐾)
63	2, 7, 56, 57, 58, 59, 60, 61	catlid 17392	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
64	62, 63	eqtr4d 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾))
65	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
66		simpr1 1193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧 ∈ 𝐵)
67		eqid 2738	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
68		simpr2 1194	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤 ∈ 𝐵)
69		simpr3 1195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
70	6, 27, 8, 65, 66, 67, 68, 69	catlid 17392	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼‘𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
71	6, 27, 8, 65, 66, 67, 68, 69	catrid 17393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼‘𝑧)) = 𝑓)
72	70, 71	eqtr4d 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼‘𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼‘𝑧)))
73	64, 72	opeq12d 4812	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨(𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼‘𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓)⟩ = ⟨(((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾), (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼‘𝑧))⟩)
74		eqid 2738	. . . . . . . 8 ⊢ (comp‘(𝐶 ×_c 𝐷)) = (comp‘(𝐶 ×_c 𝐷))
75	2, 7, 56, 57, 58	catidcl 17391	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
76	6, 27, 8, 65, 68	catidcl 17391	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
77	14, 2, 6, 7, 27, 58, 66, 58, 68, 59, 67, 74, 60, 68, 75, 69, 61, 76	xpcco2 17904	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝐾, (𝐼‘𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩) = ⟨(𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼‘𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓)⟩)
78	36	3ad2antr1 1187	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
79	2, 7, 56, 57, 60	catidcl 17391	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌))
80	14, 2, 6, 7, 27, 58, 66, 60, 66, 59, 67, 74, 60, 68, 61, 78, 79, 69	xpcco2 17904	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼‘𝑧)⟩) = ⟨(((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾), (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼‘𝑧))⟩)
81	73, 77, 80	3eqtr4d 2788	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝐾, (𝐼‘𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩) = (⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼‘𝑧)⟩))
82	81	fveq2d 6778	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘(⟨𝐾, (𝐼‘𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼‘𝑧)⟩)))
83		eqid 2738	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
84	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1^st ‘𝐹)((𝐶 ×_c 𝐷) Func 𝐸)(2^nd ‘𝐹))
85	23	3ad2antr1 1187	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
86	58, 68	opelxpd 5627	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
87	60, 68	opelxpd 5627	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
88	75, 69	opelxpd 5627	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
89	14, 2, 6, 7, 27, 58, 66, 58, 68, 16	xpchom2 17903	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
90	88, 89	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑋, 𝑤⟩))
91	61, 76	opelxpd 5627	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
92	14, 2, 6, 7, 27, 58, 68, 60, 68, 16	xpchom2 17903	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
93	91, 92	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑤)⟩ ∈ (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩))
94	15, 16, 74, 83, 84, 85, 86, 87, 90, 93	funcco 17586	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘(⟨𝐾, (𝐼‘𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = (((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)))
95	25	3ad2antr1 1187	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
96	61, 78	opelxpd 5627	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
97	14, 2, 6, 7, 27, 58, 66, 60, 66, 16	xpchom2 17903	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
98	96, 97	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩))
99	79, 69	opelxpd 5627	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
100	14, 2, 6, 7, 27, 60, 66, 60, 68, 16	xpchom2 17903	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
101	99, 100	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩))
102	15, 16, 74, 83, 84, 85, 95, 87, 98, 101	funcco 17586	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼‘𝑧)⟩)) = (((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩)))
103	82, 94, 102	3eqtr3d 2786	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = (((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩)))
104	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸))
105	1, 2, 57, 65, 104, 6, 58, 40, 66	curf11 17944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1^st ‘𝐹)𝑧))
106	105, 42	eqtrdi 2794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩))
107	1, 2, 57, 65, 104, 6, 58, 40, 68	curf11 17944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤) = (𝑋(1^st ‘𝐹)𝑤))
108		df-ov 7278	. . . . . . . 8 ⊢ (𝑋(1^st ‘𝐹)𝑤) = ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)
109	107, 108	eqtrdi 2794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤) = ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩))
110	106, 109	opeq12d 4812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩ = ⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)⟩)
111	1, 2, 57, 65, 104, 6, 60, 44, 68	curf11 17944	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤) = (𝑌(1^st ‘𝐹)𝑤))
112		df-ov 7278	. . . . . . 7 ⊢ (𝑌(1^st ‘𝐹)𝑤) = ((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩)
113	111, 112	eqtrdi 2794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤) = ((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))
114	110, 113	oveq12d 7293	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤)) = (⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩)))
115	1, 2, 57, 65, 104, 6, 7, 8, 58, 60, 61, 12, 68	curf2val 17948	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)(𝐼‘𝑤)))
116		df-ov 7278	. . . . . 6 ⊢ (𝐾(⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)(𝐼‘𝑤)) = ((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩)
117	115, 116	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = ((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩))
118	1, 2, 57, 65, 104, 6, 58, 40, 66, 27, 56, 68, 69	curf12 17945	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)𝑓))
119		df-ov 7278	. . . . . 6 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
120	118, 119	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩))
121	114, 117, 120	oveq123d 7296	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿‘𝑤)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)))
122	1, 2, 57, 65, 104, 6, 60, 44, 66	curf11 17944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1^st ‘𝐹)𝑧))
123	122, 46	eqtrdi 2794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩))
124	106, 123	opeq12d 4812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩ = ⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)⟩)
125	124, 113	oveq12d 7293	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤)) = (⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩)))
126	1, 2, 57, 65, 104, 6, 60, 44, 66, 27, 56, 68, 69	curf12 17945	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)𝑓))
127		df-ov 7278	. . . . . 6 ⊢ (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
128	126, 127	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓) = ((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩))
129	1, 2, 57, 65, 104, 6, 7, 8, 58, 60, 61, 12, 66	curf2val 17948	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))
130		df-ov 7278	. . . . . 6 ⊢ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩)
131	129, 130	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩))
132	125, 128, 131	oveq123d 7296	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧)) = (((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩), ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩)))
133	103, 121, 132	3eqtr4d 2788	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿‘𝑤)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧)))
134	133	ralrimivvva 3127	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿‘𝑤)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧)))
135		curf2.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
136	1, 2, 3, 4, 5, 6, 9, 40	curf1cl 17946	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
137	1, 2, 3, 4, 5, 6, 10, 44	curf1cl 17946	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
138	135, 6, 27, 17, 83, 136, 137	isnat2 17664	. 2 ⊢ (𝜑 → (𝐿 ∈ (((1^st ‘𝐺)‘𝑋)𝑁((1^st ‘𝐺)‘𝑌)) ↔ (𝐿 ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿‘𝑤)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2^nd ‘((1^st ‘𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧)))))
139	55, 134, 138	mpbir2and 710	1 ⊢ (𝜑 → 𝐿 ∈ (((1^st ‘𝐺)‘𝑋)𝑁((1^st ‘𝐺)‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 Rel wrel 5594 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Xcixp 8685 Basecbs 16912 Hom chom 16973 compcco 16974 Catccat 17373 Idccid 17374 Func cfunc 17569 Nat cnat 17657 ×_c cxpc 17885 curry_F ccurf 17928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-cat 17377 df-cid 17378 df-func 17573 df-nat 17659 df-xpc 17889 df-curf 17932

This theorem is referenced by: curfcl 17950

W3C validator