Theorem evlfcl 18279

Description: The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐶⟶𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlfcl.e	⊢ 𝐸 = (𝐶 evalF 𝐷)
evlfcl.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
evlfcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
evlfcl.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
evlfcl	⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Proof of Theorem evlfcl

Dummy variables 𝑓 𝑎 𝑔 ℎ 𝑚 𝑛 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfcl.e	. . . . 5 ⊢ 𝐸 = (𝐶 evalF 𝐷)
2		evlfcl.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		evlfcl.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		eqid 2735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2735	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2735	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
7		eqid 2735	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8	1, 2, 3, 4, 5, 6, 7	evlfval 18274	. . . 4 ⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩)
9		ovex 7464	. . . . . 6 ⊢ (𝐶 Func 𝐷) ∈ V
10		fvex 6920	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
11	9, 10	mpoex 8103	. . . . 5 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) ∈ V
12	9, 10	xpex 7772	. . . . . 6 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∈ V
13	12, 12	mpoex 8103	. . . . 5 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) ∈ V
14	11, 13	opelvv 5729	. . . 4 ⊢ ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩ ∈ (V × V)
15	8, 14	eqeltrdi 2847	. . 3 ⊢ (𝜑 → 𝐸 ∈ (V × V))
16		1st2nd2 8052	. . 3 ⊢ (𝐸 ∈ (V × V) → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
18		eqid 2735	. . . . 5 ⊢ (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
19		evlfcl.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
20	19	fucbas 18016	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
21	18, 20, 4	xpcbas 18234	. . . 4 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) = (Base‘(𝑄 ×c 𝐶))
22		eqid 2735	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
23		eqid 2735	. . . 4 ⊢ (Hom ‘(𝑄 ×c 𝐶)) = (Hom ‘(𝑄 ×c 𝐶))
24		eqid 2735	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
25		eqid 2735	. . . 4 ⊢ (Id‘(𝑄 ×c 𝐶)) = (Id‘(𝑄 ×c 𝐶))
26		eqid 2735	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
27		eqid 2735	. . . 4 ⊢ (comp‘(𝑄 ×c 𝐶)) = (comp‘(𝑄 ×c 𝐶))
28	19, 2, 3	fuccat 18027	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
29	18, 28, 2	xpccat 18246	. . . 4 ⊢ (𝜑 → (𝑄 ×c 𝐶) ∈ Cat)
30		relfunc 17913	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
31		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
32		1st2ndbr 8066	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
33	30, 31, 32	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
34	4, 22, 33	funcf1 17917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelcdmda 7104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
36	35	ralrimiva 3144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
37	36	ralrimiva 3144	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
38		eqid 2735	. . . . . . 7 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥))
39	38	fmpo 8092	. . . . . 6 ⊢ (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷) ↔ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
40	37, 39	sylib 218	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
41	11, 13	op1std 8023	. . . . . . 7 ⊢ (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩ → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)))
42	8, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)))
43	42	feq1d 6721	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐸):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷) ↔ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷)))
44	40, 43	mpbird 257	. . . 4 ⊢ (𝜑 → (1st ‘𝐸):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
45		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))
46		ovex 7464	. . . . . . . . 9 ⊢ (𝑚(𝐶 Nat 𝐷)𝑛) ∈ V
47		ovex 7464	. . . . . . . . 9 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
48	46, 47	mpoex 8103	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))) ∈ V
49	48	csbex 5317	. . . . . . 7 ⊢ ⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))) ∈ V
50	49	csbex 5317	. . . . . 6 ⊢ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))) ∈ V
51	45, 50	fnmpoi 8094	. . . . 5 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶)))
52	11, 13	op2ndd 8024	. . . . . . 7 ⊢ (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩ → (2nd ‘𝐸) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))))
53	8, 52	syl 17	. . . . . 6 ⊢ (𝜑 → (2nd ‘𝐸) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))))
54	53	fneq1d 6662	. . . . 5 ⊢ (𝜑 → ((2nd ‘𝐸) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶))) ↔ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶)))))
55	51, 54	mpbiri 258	. . . 4 ⊢ (𝜑 → (2nd ‘𝐸) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶))))
56	3	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
57	56	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝐷 ∈ Cat)
58		simplrl 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐶 Func 𝐷))
59	30, 58, 32	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
60	4, 22, 59	funcf1 17917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
61	60	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
62		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑢 ∈ (Base‘𝐶))
63	62	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑢 ∈ (Base‘𝐶))
64	61, 63	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st ‘𝑓)‘𝑢) ∈ (Base‘𝐷))
65		simplrr 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑣 ∈ (Base‘𝐶))
66	61, 65	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st ‘𝑓)‘𝑣) ∈ (Base‘𝐷))
67		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑔 ∈ (𝐶 Func 𝐷))
68		1st2ndbr 8066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑔)(𝐶 Func 𝐷)(2nd ‘𝑔))
69	30, 67, 68	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st ‘𝑔)(𝐶 Func 𝐷)(2nd ‘𝑔))
70	4, 22, 69	funcf1 17917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
71	70	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
72	71, 65	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st ‘𝑔)‘𝑣) ∈ (Base‘𝐷))
73		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑣 ∈ (Base‘𝐶))
74	4, 5, 24, 59, 62, 73	funcf2 17919	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑢(2nd ‘𝑓)𝑣):(𝑢(Hom ‘𝐶)𝑣)⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑓)‘𝑣)))
75	74	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (𝑢(2nd ‘𝑓)𝑣):(𝑢(Hom ‘𝐶)𝑣)⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑓)‘𝑣)))
76		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))
77	75, 76	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((𝑢(2nd ‘𝑓)𝑣)‘ℎ) ∈ (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑓)‘𝑣)))
78		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
79	7, 78	nat1st2nd 18006	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑎 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
80	7, 79, 4, 24, 65	natcl 18008	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (𝑎‘𝑣) ∈ (((1st ‘𝑓)‘𝑣)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
81	22, 24, 6, 57, 64, 66, 72, 77, 80	catcocl 17730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ℎ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ)) ∈ (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
82	81	ralrimivva 3200	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀ℎ ∈ (𝑢(Hom ‘𝐶)𝑣)((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ)) ∈ (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
83		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ))) = (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ)))
84	83	fmpo 8092	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀ℎ ∈ (𝑢(Hom ‘𝐶)𝑣)((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ)) ∈ (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)) ↔ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
85	82, 84	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
86	2	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
87		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩)
88	1, 86, 56, 4, 5, 6, 7, 58, 67, 62, 73, 87	evlf2 18275	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩) = (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ))))
89	88	feq1d 6721	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)) ↔ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ℎ ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎‘𝑣)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑣)⟩(comp‘𝐷)((1st ‘𝑔)‘𝑣))((𝑢(2nd ‘𝑓)𝑣)‘ℎ))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣))))
90	85, 89	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
91	19, 7	fuchom 18017	. . . . . . . . . . . . 13 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
92	18, 20, 4, 91, 5, 58, 62, 67, 73, 23	xpchom2 18242	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = ((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣)))
93	1, 86, 56, 4, 58, 62	evlf1 18277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝑓)‘𝑢))
94	1, 86, 56, 4, 67, 73	evlf1 18277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑔(1st ‘𝐸)𝑣) = ((1st ‘𝑔)‘𝑣))
95	93, 94	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)) = (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣)))
96	92, 95	feq23d 6732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)) ↔ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑔)‘𝑣))))
97	90, 96	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
98	97	ralrimivva 3200	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
99	98	ralrimivva 3200	. . . . . . . 8 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
100		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(2nd ‘𝐸)𝑦) = (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
101		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
102		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩))
103		df-ov 7434	. . . . . . . . . . . . . 14 ⊢ (𝑔(1st ‘𝐸)𝑣) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩)
104	102, 103	eqtr4di 2793	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = (𝑔(1st ‘𝐸)𝑣))
105	104	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)) = (((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
106	100, 101, 105	feq123d 6726	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)) ↔ (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣))))
107	106	ralxp 5855	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
108		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
109		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
110		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩))
111		df-ov 7434	. . . . . . . . . . . . . 14 ⊢ (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩)
112	110, 111	eqtr4di 2793	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = (𝑓(1st ‘𝐸)𝑢))
113	112	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)) = ((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
114	108, 109, 113	feq123d 6726	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)) ↔ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣))))
115	114	2ralbidv 3219	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣))))
116	107, 115	bitrid 283	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣))))
117	116	ralxp 5855	. . . . . . . 8 ⊢ (∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)) ↔ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st ‘𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st ‘𝐸)𝑣)))
118	99, 117	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)))
119	118	r19.21bi 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → ∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)))
120	119	r19.21bi 3249	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → (𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)))
121	120	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))) → (𝑥(2nd ‘𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st ‘𝐸)‘𝑥)(Hom ‘𝐷)((1st ‘𝐸)‘𝑦)))
122	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑄 ∈ Cat)
123	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
124		eqid 2735	. . . . . . . . . . 11 ⊢ (Id‘𝑄) = (Id‘𝑄)
125		eqid 2735	. . . . . . . . . . 11 ⊢ (Id‘𝐶) = (Id‘𝐶)
126		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐶 Func 𝐷))
127		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑢 ∈ (Base‘𝐶))
128	18, 122, 123, 20, 4, 124, 125, 25, 126, 127	xpcid 18245	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩) = ⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
129	128	fveq2d 6911	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩))
130		df-ov 7434	. . . . . . . . 9 ⊢ (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)) = ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
131	129, 130	eqtr4di 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)))
132	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
133		eqid 2735	. . . . . . . . 9 ⊢ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)
134	20, 91, 124, 122, 126	catidcl 17727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) ∈ (𝑓(𝐶 Nat 𝐷)𝑓))
135	4, 5, 125, 123, 127	catidcl 17727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐶)‘𝑢) ∈ (𝑢(Hom ‘𝐶)𝑢))
136	1, 123, 132, 4, 5, 6, 7, 126, 126, 127, 127, 133, 134, 135	evlf2val 18276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)) = ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑢)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑢))((𝑢(2nd ‘𝑓)𝑢)‘((Id‘𝐶)‘𝑢))))
137	30, 126, 32	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
138	4, 22, 137	funcf1 17917	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
139	138, 127	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((1st ‘𝑓)‘𝑢) ∈ (Base‘𝐷))
140	22, 24, 26, 132, 139	catidcl 17727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)) ∈ (((1st ‘𝑓)‘𝑢)(Hom ‘𝐷)((1st ‘𝑓)‘𝑢)))
141	22, 24, 26, 132, 139, 6, 139, 140	catlid 17728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝐷)‘((1st ‘𝑓)‘𝑢))(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑢)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑢))((Id‘𝐷)‘((1st ‘𝑓)‘𝑢))) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
142	19, 124, 26, 126	fucid 18028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) = ((Id‘𝐷) ∘ (1st ‘𝑓)))
143	142	fveq1d 6909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢))
144		fvco3 7008	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑢 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
145	138, 127, 144	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
146	143, 145	eqtrd 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
147	4, 125, 26, 137, 127	funcid 17921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((𝑢(2nd ‘𝑓)𝑢)‘((Id‘𝐶)‘𝑢)) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
148	146, 147	oveq12d 7449	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑢)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑢))((𝑢(2nd ‘𝑓)𝑢)‘((Id‘𝐶)‘𝑢))) = (((Id‘𝐷)‘((1st ‘𝑓)‘𝑢))(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑢)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑢))((Id‘𝐷)‘((1st ‘𝑓)‘𝑢))))
149	1, 123, 132, 4, 126, 127	evlf1 18277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝑓)‘𝑢))
150	149	fveq2d 6911	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
151	141, 148, 150	3eqtr4d 2785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st ‘𝑓)‘𝑢), ((1st ‘𝑓)‘𝑢)⟩(comp‘𝐷)((1st ‘𝑓)‘𝑢))((𝑢(2nd ‘𝑓)𝑢)‘((Id‘𝐶)‘𝑢))) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
152	131, 136, 151	3eqtrd 2779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
153	152	ralrimivva 3200	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
154		id 22	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → 𝑥 = ⟨𝑓, 𝑢⟩)
155	154, 154	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)𝑥) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩))
156		fveq2 6907	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘(𝑄 ×c 𝐶))‘𝑥) = ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩))
157	155, 156	fveq12d 6914	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((𝑥(2nd ‘𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)))
158	112	fveq2d 6911	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
159	157, 158	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (((𝑥(2nd ‘𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)) ↔ ((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢))))
160	159	ralxp 5855	. . . . . 6 ⊢ (∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))((𝑥(2nd ‘𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)) ↔ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)((⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
161	153, 160	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))((𝑥(2nd ‘𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)))
162	161	r19.21bi 3249	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → ((𝑥(2nd ‘𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)))
163	2	3ad2ant1 1132	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝐶 ∈ Cat)
164	3	3ad2ant1 1132	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝐷 ∈ Cat)
165		simp21 1205	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
166		1st2nd2 8052	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
167	165, 166	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
168	167, 165	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
169		opelxp 5725	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑥) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑥) ∈ (Base‘𝐶)))
170	168, 169	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝑥) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑥) ∈ (Base‘𝐶)))
171		simp22 1206	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
172		1st2nd2 8052	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
173	171, 172	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
174	173, 171	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
175		opelxp 5725	. . . . . . 7 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑦) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑦) ∈ (Base‘𝐶)))
176	174, 175	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝑦) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑦) ∈ (Base‘𝐶)))
177		simp23 1207	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
178		1st2nd2 8052	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
179	177, 178	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
180	179, 177	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
181		opelxp 5725	. . . . . . 7 ⊢ (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑧) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑧) ∈ (Base‘𝐶)))
182	180, 181	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝑧) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑧) ∈ (Base‘𝐶)))
183		simp3l 1200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦))
184	18, 21, 91, 5, 23, 165, 171	xpchom 18236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
185	183, 184	eleqtrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 ∈ (((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
186		1st2nd2 8052	. . . . . . . . 9 ⊢ (𝑓 ∈ (((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
187	185, 186	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
188	187, 185	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩ ∈ (((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
189		opelxp 5725	. . . . . . 7 ⊢ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩ ∈ (((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) ↔ ((1st ‘𝑓) ∈ ((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) ∧ (2nd ‘𝑓) ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
190	188, 189	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝑓) ∈ ((1st ‘𝑥)(𝐶 Nat 𝐷)(1st ‘𝑦)) ∧ (2nd ‘𝑓) ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
191		simp3r 1201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))
192	18, 21, 91, 5, 23, 171, 177	xpchom 18236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧) = (((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
193	191, 192	eleqtrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 ∈ (((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
194		1st2nd2 8052	. . . . . . . . 9 ⊢ (𝑔 ∈ (((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
195	193, 194	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
196	195, 193	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∈ (((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
197		opelxp 5725	. . . . . . 7 ⊢ (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∈ (((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) ↔ ((1st ‘𝑔) ∈ ((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) ∧ (2nd ‘𝑔) ∈ ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
198	196, 197	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝑔) ∈ ((1st ‘𝑦)(𝐶 Nat 𝐷)(1st ‘𝑧)) ∧ (2nd ‘𝑔) ∈ ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
199	1, 19, 163, 164, 7, 170, 176, 182, 190, 198	evlfcllem 18278	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘(⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)) = (((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)(⟨((1st ‘𝐸)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩), ((1st ‘𝐸)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)⟩(comp‘𝐷)((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)‘⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)))
200	167, 179	oveq12d 7449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(2nd ‘𝐸)𝑧) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
201	167, 173	opeq12d 4886	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨𝑥, 𝑦⟩ = ⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩)
202	201, 179	oveq12d 7449	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧) = (⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
203	202, 195, 187	oveq123d 7452	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
204	200, 203	fveq12d 6914	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝐸)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓)) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘(⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)))
205	167	fveq2d 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝐸)‘𝑥) = ((1st ‘𝐸)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
206	173	fveq2d 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝐸)‘𝑦) = ((1st ‘𝐸)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
207	205, 206	opeq12d 4886	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨((1st ‘𝐸)‘𝑥), ((1st ‘𝐸)‘𝑦)⟩ = ⟨((1st ‘𝐸)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩), ((1st ‘𝐸)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)⟩)
208	179	fveq2d 6911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
209	207, 208	oveq12d 7449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (⟨((1st ‘𝐸)‘𝑥), ((1st ‘𝐸)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐸)‘𝑧)) = (⟨((1st ‘𝐸)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩), ((1st ‘𝐸)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)⟩(comp‘𝐷)((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)))
210	173, 179	oveq12d 7449	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑦(2nd ‘𝐸)𝑧) = (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
211	210, 195	fveq12d 6914	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑦(2nd ‘𝐸)𝑧)‘𝑔) = ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
212	167, 173	oveq12d 7449	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(2nd ‘𝐸)𝑦) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
213	212, 187	fveq12d 6914	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝐸)𝑦)‘𝑓) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)‘⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
214	209, 211, 213	oveq123d 7452	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (((𝑦(2nd ‘𝐸)𝑧)‘𝑔)(⟨((1st ‘𝐸)‘𝑥), ((1st ‘𝐸)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐸)‘𝑧))((𝑥(2nd ‘𝐸)𝑦)‘𝑓)) = (((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝐸)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)(⟨((1st ‘𝐸)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩), ((1st ‘𝐸)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)⟩(comp‘𝐷)((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝐸)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)‘⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)))
215	199, 204, 214	3eqtr4d 2785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝐸)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓)) = (((𝑦(2nd ‘𝐸)𝑧)‘𝑔)(⟨((1st ‘𝐸)‘𝑥), ((1st ‘𝐸)‘𝑦)⟩(comp‘𝐷)((1st ‘𝐸)‘𝑧))((𝑥(2nd ‘𝐸)𝑦)‘𝑓)))
216	21, 22, 23, 24, 25, 26, 27, 6, 29, 3, 44, 55, 121, 162, 215	isfuncd 17916	. . 3 ⊢ (𝜑 → (1st ‘𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd ‘𝐸))
217		df-br 5149	. . 3 ⊢ ((1st ‘𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd ‘𝐸) ↔ ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
218	216, 217	sylib 218	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
219	17, 218	eqeltrd 2839	1 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  ⦋csb 3908  ⟨cop 4637   class class class wbr 5148   × cxp 5687   ∘ ccom 5693  Rel wrel 5694   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8011  2^nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710   Func cfunc 17905   Nat cnat 17996   FuncCat cfuc 17997   ×_c cxpc 18224   eval_F cevlf 18266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-func 17909  df-nat 17998  df-fuc 17999  df-xpc 18228  df-evlf 18270

This theorem is referenced by:  uncfcl  18292  uncf1  18293  uncf2  18294  yonedalem1  18329