curf2cl - Metamath Proof Explorer

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Theorem curf2cl 16871

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 16869	. . 3 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
14		eqid 2622	. . . . . . . . . 10 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
15	14, 2, 6	xpcbas 16818	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
16		eqid 2622	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
17		eqid 2622	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
18		relfunc 16522	. . . . . . . . . . 11 ⊢ Rel ((𝐶 ×_c 𝐷) Func 𝐸)
19		1st2ndbr 7217	. . . . . . . . . . 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 695	. . . . . . . . . 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 5148	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
23	9, 22	sylan 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24		opelxpi 5148	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
25	10, 24	sylan 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
26	15, 16, 17, 21, 23, 25	funcf2 16528	. . . . . . . 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 2622	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
28	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐴)
29		simpr 477	. . . . . . . . . 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 16826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
32	31	feq2d 6031	. . . . . . . 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 222	. . . . . . 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 16343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
37	33, 34, 36	fovrnd 6806	. . . . . 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 2622	. . . . . . . . 9 ⊢ ((1^st ‘𝐺)‘𝑋) = ((1^st ‘𝐺)‘𝑋)
41	1, 2, 38, 35, 39, 6, 28, 40, 29	curf11 16866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1^st ‘𝐹)𝑧))
42		df-ov 6653	. . . . . . . 8 ⊢ (𝑋(1^st ‘𝐹)𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩)
43	41, 42	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩))
44		eqid 2622	. . . . . . . . 9 ⊢ ((1^st ‘𝐺)‘𝑌) = ((1^st ‘𝐺)‘𝑌)
45	1, 2, 38, 35, 39, 6, 30, 44, 29	curf11 16866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1^st ‘𝐹)𝑧))
46		df-ov 6653	. . . . . . . 8 ⊢ (𝑌(1^st ‘𝐹)𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩)
47	45, 46	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩))
48	43, 47	oveq12d 6668	. . . . . 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 2704	. . . . 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 2966	. . . 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		fvex 6201	. . . . . 6 ⊢ (Base‘𝐷) ∈ V
52	6, 51	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
53		mptelixpg 7945	. . . . 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))))
54	52, 53	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)))
55	50, 54	sylibr 224	. . 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)))
56	13, 55	eqeltrd 2701	. 2 ⊢ (𝜑 → 𝐿 ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧)))
57		eqid 2622	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
58	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
59	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋 ∈ 𝐴)
60		eqid 2622	. . . . . . . . . 10 ⊢ (comp‘𝐶) = (comp‘𝐶)
61	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌 ∈ 𝐴)
62	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌))
63	2, 7, 57, 58, 59, 60, 61, 62	catrid 16345	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐾)
64	2, 7, 57, 58, 59, 60, 61, 62	catlid 16344	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
65	63, 64	eqtr4d 2659	. . . . . . . 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))
66	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
67		simpr1 1067	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧 ∈ 𝐵)
68		eqid 2622	. . . . . . . . . 10 ⊢ (comp‘𝐷) = (comp‘𝐷)
69		simpr2 1068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤 ∈ 𝐵)
70		simpr3 1069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
71	6, 27, 8, 66, 67, 68, 69, 70	catlid 16344	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼‘𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
72	6, 27, 8, 66, 67, 68, 69, 70	catrid 16345	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼‘𝑧)) = 𝑓)
73	71, 72	eqtr4d 2659	. . . . . . . 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)))
74	65, 73	opeq12d 4410	. . . . . . 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))⟩)
75		eqid 2622	. . . . . . . 8 ⊢ (comp‘(𝐶 ×_c 𝐷)) = (comp‘(𝐶 ×_c 𝐷))
76	2, 7, 57, 58, 59	catidcl 16343	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
77	6, 27, 8, 66, 69	catidcl 16343	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
78	14, 2, 6, 7, 27, 59, 67, 59, 69, 60, 68, 75, 61, 69, 76, 70, 62, 77	xpcco2 16827	. . . . . . 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)⟩)
79	36	3ad2antr1 1226	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
80	2, 7, 57, 58, 61	catidcl 16343	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌))
81	14, 2, 6, 7, 27, 59, 67, 61, 67, 60, 68, 75, 61, 69, 62, 79, 80, 70	xpcco2 16827	. . . . . . 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))⟩)
82	74, 78, 81	3eqtr4d 2666	. . . . . 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)⟩))
83	82	fveq2d 6195	. . . . 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)⟩)))
84		eqid 2622	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
85	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))
86	23	3ad2antr1 1226	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
87		opelxpi 5148	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
88	59, 69, 87	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
89		opelxpi 5148	. . . . . . 7 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
90	61, 69, 89	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
91		opelxpi 5148	. . . . . . . 8 ⊢ ((((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
92	76, 70, 91	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
93	14, 2, 6, 7, 27, 59, 67, 59, 69, 16	xpchom2 16826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
94	92, 93	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑋, 𝑤⟩))
95		opelxpi 5148	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝐾, (𝐼‘𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
96	62, 77, 95	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
97	14, 2, 6, 7, 27, 59, 69, 61, 69, 16	xpchom2 16826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
98	96, 97	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑤)⟩ ∈ (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩))
99	15, 16, 75, 84, 85, 86, 88, 90, 94, 98	funcco 16531	. . . . 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⟩)))
100	25	3ad2antr1 1226	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
101		opelxpi 5148	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) → ⟨𝐾, (𝐼‘𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
102	62, 79, 101	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
103	14, 2, 6, 7, 27, 59, 67, 61, 67, 16	xpchom2 16826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
104	102, 103	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼‘𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑧⟩))
105		opelxpi 5148	. . . . . . . 8 ⊢ ((((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
106	80, 70, 105	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
107	14, 2, 6, 7, 27, 61, 67, 61, 69, 16	xpchom2 16826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
108	106, 107	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×_c 𝐷))⟨𝑌, 𝑤⟩))
109	15, 16, 75, 84, 85, 86, 100, 90, 104, 108	funcco 16531	. . . . 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)⟩)))
110	83, 99, 109	3eqtr3d 2664	. . . 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)⟩)))
111	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×_c 𝐷) Func 𝐸))
112	1, 2, 58, 66, 111, 6, 59, 40, 67	curf11 16866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1^st ‘𝐹)𝑧))
113	112, 42	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑋, 𝑧⟩))
114	1, 2, 58, 66, 111, 6, 59, 40, 69	curf11 16866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤) = (𝑋(1^st ‘𝐹)𝑤))
115		df-ov 6653	. . . . . . . 8 ⊢ (𝑋(1^st ‘𝐹)𝑤) = ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩)
116	114, 115	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑤) = ((1^st ‘𝐹)‘⟨𝑋, 𝑤⟩))
117	113, 116	opeq12d 4410	. . . . . 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⟩)⟩)
118	1, 2, 58, 66, 111, 6, 61, 44, 69	curf11 16866	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤) = (𝑌(1^st ‘𝐹)𝑤))
119		df-ov 6653	. . . . . . 7 ⊢ (𝑌(1^st ‘𝐹)𝑤) = ((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩)
120	118, 119	syl6eq 2672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑤) = ((1^st ‘𝐹)‘⟨𝑌, 𝑤⟩))
121	117, 120	oveq12d 6668	. . . . 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⟩)))
122	1, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 69	curf2val 16870	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)(𝐼‘𝑤)))
123		df-ov 6653	. . . . . 6 ⊢ (𝐾(⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)(𝐼‘𝑤)) = ((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩)
124	122, 123	syl6eq 2672	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = ((⟨𝑋, 𝑤⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼‘𝑤)⟩))
125	1, 2, 58, 66, 111, 6, 59, 40, 67, 27, 57, 69, 70	curf12 16867	. . . . . 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))
126		df-ov 6653	. . . . . 6 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
127	125, 126	syl6eq 2672	. . . . 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⟩))
128	121, 124, 127	oveq123d 6671	. . . 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⟩)))
129	1, 2, 58, 66, 111, 6, 61, 44, 67	curf11 16866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1^st ‘𝐹)𝑧))
130	129, 46	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1^st ‘((1^st ‘𝐺)‘𝑌))‘𝑧) = ((1^st ‘𝐹)‘⟨𝑌, 𝑧⟩))
131	113, 130	opeq12d 4410	. . . . . 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⟩)⟩)
132	131, 120	oveq12d 6668	. . . . 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⟩)))
133	1, 2, 58, 66, 111, 6, 61, 44, 67, 27, 57, 69, 70	curf12 16867	. . . . . 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))
134		df-ov 6653	. . . . . 6 ⊢ (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
135	133, 134	syl6eq 2672	. . . . 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⟩))
136	1, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 67	curf2val 16870	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))
137		df-ov 6653	. . . . . 6 ⊢ (𝐾(⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩)
138	136, 137	syl6eq 2672	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = ((⟨𝑋, 𝑧⟩(2^nd ‘𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼‘𝑧)⟩))
139	132, 135, 138	oveq123d 6671	. . . 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)⟩)))
140	110, 128, 139	3eqtr4d 2666	. . 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)))
141	140	ralrimivvva 2972	. 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)))
142		curf2.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
143	1, 2, 3, 4, 5, 6, 9, 40	curf1cl 16868	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
144	1, 2, 3, 4, 5, 6, 10, 44	curf1cl 16868	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
145	142, 6, 27, 17, 84, 143, 144	isnat2 16608	. 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)))))
146	56, 141, 145	mpbir2and 957	1 ⊢ (𝜑 → 𝐿 ∈ (((1^st ‘𝐺)‘𝑋)𝑁((1^st ‘𝐺)‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 Rel wrel 5119 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 Xcixp 7908 Basecbs 15857 Hom chom 15952 compcco 15953 Catccat 16325 Idccid 16326 Func cfunc 16514 Nat cnat 16601 ×_c cxpc 16808 curry_F ccurf 16850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-cat 16329 df-cid 16330 df-func 16518 df-nat 16603 df-xpc 16812 df-curf 16854

This theorem is referenced by: curfcl 16872

W3C validator