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Theorem evlfval 18274

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

**Hypotheses**
Ref	Expression
evlfval.e	⊢ 𝐸 = (𝐶 eval_F 𝐷)
evlfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
evlfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
evlfval.b	⊢ 𝐵 = (Base‘𝐶)
evlfval.h	⊢ 𝐻 = (Hom ‘𝐶)
evlfval.o	⊢ · = (comp‘𝐷)
evlfval.n	⊢ 𝑁 = (𝐶 Nat 𝐷)

**Assertion**
Ref	Expression
evlfval	⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)

Distinct variable groups: 𝑓,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦,𝐶 𝐷,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦 𝑔,𝐻,𝑚,𝑛,𝑥,𝑦 𝑁,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦 𝜑,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦 · ,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦 𝑥,𝐵,𝑦 Allowed substitution hints: 𝐵(𝑓,𝑔,𝑚,𝑛,𝑎) · (𝑓) 𝐸(𝑥,𝑦,𝑓,𝑔,𝑚,𝑛,𝑎) 𝐻(𝑓,𝑎) 𝑁(𝑓)

Proof of Theorem evlfval Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfval.e	. 2 ⊢ 𝐸 = (𝐶 eval_F 𝐷)
2		df-evlf 18270	. . . 4 ⊢ eval_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)
3	2	a1i 11	. . 3 ⊢ (𝜑 → eval_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩))
4		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	oveq12d 7430	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	4	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Base‘𝑐) = (Base‘𝐶))
8		evlfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
9	7, 8	eqtr4di 2816	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Base‘𝑐) = 𝐵)
10		eqidd 2764	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((1^st ‘𝑓)‘𝑥) = ((1^st ‘𝑓)‘𝑥))
11	6, 9, 10	mpoeq123dv 7487	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1^st ‘𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)))
12	6, 9	xpeq12d 5694	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((𝑐 Func 𝑑) × (Base‘𝑐)) = ((𝐶 Func 𝐷) × 𝐵))
13	4, 5	oveq12d 7430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
14		evlfval.n	. . . . . . . . . 10 ⊢ 𝑁 = (𝐶 Nat 𝐷)
15	13, 14	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = 𝑁)
16	15	oveqd 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑚(𝑐 Nat 𝑑)𝑛) = (𝑚𝑁𝑛))
17	4	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Hom ‘𝑐) = (Hom ‘𝐶))
18		evlfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
19	17, 18	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Hom ‘𝑐) = 𝐻)
20	19	oveqd 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) = ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)))
21	5	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (comp‘𝑑) = (comp‘𝐷))
22		evlfval.o	. . . . . . . . . . 11 ⊢ · = (comp‘𝐷)
23	21, 22	eqtr4di 2816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (comp‘𝑑) = · )
24	23	oveqd 7429	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦))) = (⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦))))
25	24	oveqd 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔)) = ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔)))
26	16, 20, 25	mpoeq123dv 7487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))) = (𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))
27	26	csbeq2dv 3861	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))) = ⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))
28	27	csbeq2dv 3861	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))) = ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))
29	12, 12, 28	mpoeq123dv 7487	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔)))) = (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔)))))
30	11, 29	opeq12d 4847	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩ = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)
31		evlfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
32		evlfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
33		opex 5447	. . . 4 ⊢ ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩ ∈ V
34	33	a1i 11	. . 3 ⊢ (𝜑 → ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩ ∈ V)
35	3, 30, 31, 32, 34	ovmpod 7564	. 2 ⊢ (𝜑 → (𝐶 eval_F 𝐷) = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)
36	1, 35	eqtrid 2810	1 ⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩ · ((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⦋csb 3854 ⟨cop 4596 × cxp 5661 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 Hom chom 17322 compcco 17323 Catccat 17721 Func cfunc 17912 Nat cnat 18002 eval_F cevlf 18266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-pr 5406

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-evlf 18270

This theorem is referenced by: evlf2 18275 evlf1 18277 evlfcl 18279

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