Theorem evlfcl 18237

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 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2734	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
7		eqid 2734	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8	1, 2, 3, 4, 5, 6, 7	evlfval 18232	. . . 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 7446	. . . . . 6 ⊢ (𝐶 Func 𝐷) ∈ V
10		fvex 6899	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
11	9, 10	mpoex 8086	. . . . 5 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) ∈ V
12	9, 10	xpex 7755	. . . . . 6 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∈ V
13	12, 12	mpoex 8086	. . . . 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 5705	. . . 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 2841	. . 3 ⊢ (𝜑 → 𝐸 ∈ (V × V))
16		1st2nd2 8035	. . 3 ⊢ (𝐸 ∈ (V × V) → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
18		eqid 2734	. . . . 5 ⊢ (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
19		evlfcl.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
20	19	fucbas 17979	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
21	18, 20, 4	xpcbas 18193	. . . 4 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) = (Base‘(𝑄 ×c 𝐶))
22		eqid 2734	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
23		eqid 2734	. . . 4 ⊢ (Hom ‘(𝑄 ×c 𝐶)) = (Hom ‘(𝑄 ×c 𝐶))
24		eqid 2734	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
25		eqid 2734	. . . 4 ⊢ (Id‘(𝑄 ×c 𝐶)) = (Id‘(𝑄 ×c 𝐶))
26		eqid 2734	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
27		eqid 2734	. . . 4 ⊢ (comp‘(𝑄 ×c 𝐶)) = (comp‘(𝑄 ×c 𝐶))
28	19, 2, 3	fuccat 17989	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
29	18, 28, 2	xpccat 18205	. . . 4 ⊢ (𝜑 → (𝑄 ×c 𝐶) ∈ Cat)
30		relfunc 17878	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
31		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
32		1st2ndbr 8049	. . . . . . . . . . 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 17882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelcdmda 7084	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
36	35	ralrimiva 3133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
37	36	ralrimiva 3133	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
38		eqid 2734	. . . . . . 7 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥))
39	38	fmpo 8075	. . . . . 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 8006	. . . . . . 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 6700	. . . . 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 2734	. . . . . 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 7446	. . . . . . . . 9 ⊢ (𝑚(𝐶 Nat 𝐷)𝑛) ∈ V
47		ovex 7446	. . . . . . . . 9 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
48	46, 47	mpoex 8086	. . . . . . . 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 5291	. . . . . . 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 5291	. . . . . 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 8077	. . . . 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 8007	. . . . . . 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 6641	. . . . 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 776	. . . . . . . . . . . . . . . . . . 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 17882	. . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑔 ∈ (𝐶 Func 𝐷))
68		1st2ndbr 8049	. . . . . . . . . . . . . . . . . . 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 17882	. . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . 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 17884	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 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 17970	. . . . . . . . . . . . . . . 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 17972	. . . . . . . . . . . . . . 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 17699	. . . . . . . . . . . . . 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 3189	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . 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 8075	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . 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 18233	. . . . . . . . . . . . 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 6700	. . . . . . . . . . . 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 17980	. . . . . . . . . . . . 13 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
92	18, 20, 4, 91, 5, 58, 62, 67, 73, 23	xpchom2 18201	. . . . . . . . . . . 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 18235	. . . . . . . . . . . . 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 18235	. . . . . . . . . . . . 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 7431	. . . . . . . . . . . 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 6711	. . . . . . . . . . 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 3189	. . . . . . . . 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 3189	. . . . . . . 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 7421	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(2nd ‘𝐸)𝑦) = (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
101		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
102		fveq2 6886	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩))
103		df-ov 7416	. . . . . . . . . . . . . 14 ⊢ (𝑔(1st ‘𝐸)𝑣) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩)
104	102, 103	eqtr4di 2787	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = (𝑔(1st ‘𝐸)𝑣))
105	104	oveq2d 7429	. . . . . . . . . . . 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 6705	. . . . . . . . . . 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 5832	. . . . . . . . . 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 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
109		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
110		fveq2 6886	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩))
111		df-ov 7416	. . . . . . . . . . . . . 14 ⊢ (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩)
112	110, 111	eqtr4di 2787	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = (𝑓(1st ‘𝐸)𝑢))
113	112	oveq1d 7428	. . . . . . . . . . . 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 6705	. . . . . . . . . . 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 3208	. . . . . . . . . 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 5832	. . . . . . . 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 3237	. . . . . 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 3237	. . . . 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 2734	. . . . . . . . . . 11 ⊢ (Id‘𝑄) = (Id‘𝑄)
125		eqid 2734	. . . . . . . . . . 11 ⊢ (Id‘𝐶) = (Id‘𝐶)
126		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐶 Func 𝐷))
127		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑢 ∈ (Base‘𝐶))
128	18, 122, 123, 20, 4, 124, 125, 25, 126, 127	xpcid 18204	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩) = ⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
129	128	fveq2d 6890	. . . . . . . . 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 7416	. . . . . . . . 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 2787	. . . . . . . 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 2734	. . . . . . . . 9 ⊢ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)
134	20, 91, 124, 122, 126	catidcl 17696	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) ∈ (𝑓(𝐶 Nat 𝐷)𝑓))
135	4, 5, 125, 123, 127	catidcl 17696	. . . . . . . . 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 18234	. . . . . . . 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 17882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
139	138, 127	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((1st ‘𝑓)‘𝑢) ∈ (Base‘𝐷))
140	22, 24, 26, 132, 139	catidcl 17696	. . . . . . . . . 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 17697	. . . . . . . . 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 17990	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) = ((Id‘𝐷) ∘ (1st ‘𝑓)))
143	142	fveq1d 6888	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢))
144		fvco3 6988	. . . . . . . . . . . 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 2769	. . . . . . . . . 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 17886	. . . . . . . . . 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 7431	. . . . . . . . 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 18235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝑓)‘𝑢))
150	149	fveq2d 6890	. . . . . . . . 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 2779	. . . . . . . 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 2773	. . . . . . 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 3189	. . . . . 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 7431	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)𝑥) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩))
156		fveq2 6886	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘(𝑄 ×c 𝐶))‘𝑥) = ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩))
157	155, 156	fveq12d 6893	. . . . . . . 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 6890	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
159	157, 158	eqeq12d 2750	. . . . . . 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 5832	. . . . . 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 3237	. . . 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 1133	. . . . . 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 1133	. . . . . 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 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))) → 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
166		1st2nd2 8035	. . . . . . . . 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 2834	. . . . . . 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 5701	. . . . . . 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 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))) → 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
172		1st2nd2 8035	. . . . . . . . 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 2834	. . . . . . 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 5701	. . . . . . 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 1208	. . . . . . . . 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 8035	. . . . . . . . 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 2834	. . . . . . 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 5701	. . . . . . 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 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))) → 𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦))
184	18, 21, 91, 5, 23, 165, 171	xpchom 18195	. . . . . . . . . 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 2835	. . . . . . . . 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 8035	. . . . . . . . 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 2834	. . . . . . 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 5701	. . . . . . 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 1202	. . . . . . . . . 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 18195	. . . . . . . . . 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 2835	. . . . . . . . 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 8035	. . . . . . . . 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 2834	. . . . . . 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 5701	. . . . . . 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 18236	. . . . 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 7431	. . . . . 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 4861	. . . . . . . 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 7431	. . . . . . 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 7434	. . . . . 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 6893	. . . . 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 6890	. . . . . . . 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 6890	. . . . . . . 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 4861	. . . . . . 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 6890	. . . . . . 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 7431	. . . . . 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 7431	. . . . . . 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 6893	. . . . . 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 7431	. . . . . . 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 6893	. . . . . 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 7434	. . . . 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 2779	. . . 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 17881	. . 3 ⊢ (𝜑 → (1st ‘𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd ‘𝐸))
217		df-br 5124	. . 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 2833	1 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3463  ⦋csb 3879  ⟨cop 4612   class class class wbr 5123   × cxp 5663   ∘ ccom 5669  Rel wrel 5670   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994  2^nd c2nd 7995  Basecbs 17229  Hom chom 17284  compcco 17285  Catccat 17678  Idccid 17679   Func cfunc 17870   Nat cnat 17960   FuncCat cfuc 17961   ×_c cxpc 18183   eval_F cevlf 18224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17230  df-hom 17297  df-cco 17298  df-cat 17682  df-cid 17683  df-func 17874  df-nat 17962  df-fuc 17963  df-xpc 18187  df-evlf 18228

This theorem is referenced by:  uncfcl  18250  uncf1  18251  uncf2  18252  yonedalem1  18287