Theorem curfval 18185

Description: Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a	⊢ 𝐴 = (Base‘𝐶)
curfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfval.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b	⊢ 𝐵 = (Base‘𝐷)
curfval.j	⊢ 𝐽 = (Hom ‘𝐷)
curfval.1	⊢ 1 = (Id‘𝐶)
curfval.h	⊢ 𝐻 = (Hom ‘𝐶)
curfval.i	⊢ 𝐼 = (Id‘𝐷)

Assertion

Ref	Expression
curfval	⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)

Distinct variable groups:   𝑥,𝑔,𝑦,𝑧, 1   𝑥,𝐴,𝑦   𝐵,𝑔,𝑥,𝑦,𝑧   𝐶,𝑔,𝑥,𝑦,𝑧   𝐷,𝑔,𝑥,𝑦,𝑧   𝑔,𝐻,𝑦,𝑧   𝜑,𝑔,𝑥,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽,𝑥   𝑔,𝐹,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐸(𝑥)   𝐺(𝑥,𝑦,𝑧,𝑔)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curfval

Dummy variables 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfval.g	. 2 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		df-curf 18176	. . . 4 ⊢ curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌⟨(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)))))⟩)
3	2	a1i 11	. . 3 ⊢ (𝜑 → curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌⟨(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)))))⟩))
4		fvexd 6899	. . . 4 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘𝑒) ∈ V)
5		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → 𝑒 = ⟨𝐶, 𝐷⟩)
6	5	fveq2d 6888	. . . . 5 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘𝑒) = (1st ‘⟨𝐶, 𝐷⟩))
7		curfval.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		curfval.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
9		op1stg 7983	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
10	7, 8, 9	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
12	6, 11	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘𝑒) = 𝐶)
13		fvexd 6899	. . . . 5 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑒) ∈ V)
14	5	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → 𝑒 = ⟨𝐶, 𝐷⟩)
15	14	fveq2d 6888	. . . . . 6 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑒) = (2nd ‘⟨𝐶, 𝐷⟩))
16		op2ndg 7984	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
17	7, 8, 16	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
18	17	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
19	15, 18	eqtrd 2766	. . . . 5 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑒) = 𝐷)
20		simplr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
21	20	fveq2d 6888	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
22		curfval.a	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝐶)
23	21, 22	eqtr4di 2784	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐴)
24		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
25	24	fveq2d 6888	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
26		curfval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐷)
27	25, 26	eqtr4di 2784	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑑) = 𝐵)
28		simprr 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
29	28	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑓 = 𝐹)
30	29	fveq2d 6888	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (1st ‘𝑓) = (1st ‘𝐹))
31	30	oveqd 7421	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(1st ‘𝑓)𝑦) = (𝑥(1st ‘𝐹)𝑦))
32	27, 31	mpteq12dv 5232	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st ‘𝑓)𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)))
33	24	fveq2d 6888	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
34		curfval.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (Hom ‘𝐷)
35	33, 34	eqtr4di 2784	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
36	35	oveqd 7421	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐽𝑧))
37	29	fveq2d 6888	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (2nd ‘𝑓) = (2nd ‘𝐹))
38	37	oveqd 7421	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩))
39	20	fveq2d 6888	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = (Id‘𝐶))
40		curfval.1	. . . . . . . . . . . . 13 ⊢ 1 = (Id‘𝐶)
41	39, 40	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = 1 )
42	41	fveq1d 6886	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
43		eqidd 2727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑔 = 𝑔)
44	38, 42, 43	oveq123d 7425	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔) = (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))
45	36, 44	mpteq12dv 5232	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))
46	27, 27, 45	mpoeq123dv 7479	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))))
47	32, 46	opeq12d 4876	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨(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)))⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩)
48	23, 47	mpteq12dv 5232	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (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 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
49	20	fveq2d 6888	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = (Hom ‘𝐶))
50		curfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
51	49, 50	eqtr4di 2784	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = 𝐻)
52	51	oveqd 7421	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
53	37	oveqd 7421	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩) = (⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩))
54	24	fveq2d 6888	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = (Id‘𝐷))
55		curfval.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (Id‘𝐷)
56	54, 55	eqtr4di 2784	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = 𝐼)
57	56	fveq1d 6886	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑑)‘𝑧) = (𝐼‘𝑧))
58	53, 43, 57	oveq123d 7425	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)) = (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))
59	27, 58	mpteq12dv 5232	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧))) = (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))
60	52, 59	mpteq12dv 5232	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))) = (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))
61	23, 23, 60	mpoeq123dv 7479	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (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 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))))
62	48, 61	opeq12d 4876	. . . . 5 ⊢ ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨(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)))))⟩ = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
63	13, 19, 62	csbied2 3928	. . . 4 ⊢ (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → ⦋(2nd ‘𝑒) / 𝑑⦌⟨(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)))))⟩ = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
64	4, 12, 63	csbied2 3928	. . 3 ⊢ ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌⟨(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)))))⟩ = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
65		opex 5457	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
66	65	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ V)
67		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
68	67	elexd 3489	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
69		opex 5457	. . . 4 ⊢ ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩ ∈ V
70	69	a1i 11	. . 3 ⊢ (𝜑 → ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩ ∈ V)
71	3, 64, 66, 68, 70	ovmpod 7555	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
72	1, 71	eqtrid 2778	1 ⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⦋csb 3888  ⟨cop 4629   ↦ cmpt 5224  ‘cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17150  Hom chom 17214  Catccat 17614  Idccid 17615   Func cfunc 17810   ×_c cxpc 18129   curry_F ccurf 18172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-curf 18176

This theorem is referenced by:  curf1fval  18186  curf2  18191  curfcl  18194  curfpropd  18195  curfuncf  18200