Theorem evlfcl 17730

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 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2737	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
7		eqid 2737	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8	1, 2, 3, 4, 5, 6, 7	evlfval 17725	. . . 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 7246	. . . . . 6 ⊢ (𝐶 Func 𝐷) ∈ V
10		fvex 6730	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
11	9, 10	mpoex 7850	. . . . 5 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) ∈ V
12	9, 10	xpex 7538	. . . . . 6 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∈ V
13	12, 12	mpoex 7850	. . . . 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 5590	. . . 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 2846	. . 3 ⊢ (𝜑 → 𝐸 ∈ (V × V))
16		1st2nd2 7800	. . 3 ⊢ (𝐸 ∈ (V × V) → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → 𝐸 = ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩)
18		eqid 2737	. . . . 5 ⊢ (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
19		evlfcl.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
20	19	fucbas 17468	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
21	18, 20, 4	xpcbas 17685	. . . 4 ⊢ ((𝐶 Func 𝐷) × (Base‘𝐶)) = (Base‘(𝑄 ×c 𝐶))
22		eqid 2737	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
23		eqid 2737	. . . 4 ⊢ (Hom ‘(𝑄 ×c 𝐶)) = (Hom ‘(𝑄 ×c 𝐶))
24		eqid 2737	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
25		eqid 2737	. . . 4 ⊢ (Id‘(𝑄 ×c 𝐶)) = (Id‘(𝑄 ×c 𝐶))
26		eqid 2737	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
27		eqid 2737	. . . 4 ⊢ (comp‘(𝑄 ×c 𝐶)) = (comp‘(𝑄 ×c 𝐶))
28	19, 2, 3	fuccat 17479	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
29	18, 28, 2	xpccat 17697	. . . 4 ⊢ (𝜑 → (𝑄 ×c 𝐶) ∈ Cat)
30		relfunc 17368	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
31		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
32		1st2ndbr 7813	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
33	30, 31, 32	sylancr 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
34	4, 22, 33	funcf1 17372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelrnda 6904	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
36	35	ralrimiva 3105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
37	36	ralrimiva 3105	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
38		eqid 2737	. . . . . . 7 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥))
39	38	fmpo 7838	. . . . . 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 221	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
41	11, 13	op1std 7771	. . . . . . 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 6530	. . . . 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 260	. . . 4 ⊢ (𝜑 → (1st ‘𝐸):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
45		eqid 2737	. . . . . 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 7246	. . . . . . . . 9 ⊢ (𝑚(𝐶 Nat 𝐷)𝑛) ∈ V
47		ovex 7246	. . . . . . . . 9 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
48	46, 47	mpoex 7850	. . . . . . . 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 5204	. . . . . . 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 5204	. . . . . 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 7840	. . . . 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 7772	. . . . . . 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 6472	. . . . 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 261	. . . 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 484	. . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . 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 17372	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . . . . 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 7813	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st ‘𝑔)(𝐶 Func 𝐷)(2nd ‘𝑔))
69	30, 67, 68	sylancr 590	. . . . . . . . . . . . . . . . . 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 17372	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . . . . 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 17374	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . . . . 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 17458	. . . . . . . . . . . . . . . 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 17460	. . . . . . . . . . . . . . 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 17188	. . . . . . . . . . . . . 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 3112	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 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 7838	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 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 17726	. . . . . . . . . . . . 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 6530	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 17469	. . . . . . . . . . . . 13 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
92	18, 20, 4, 91, 5, 58, 62, 67, 73, 23	xpchom2 17693	. . . . . . . . . . . 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 17728	. . . . . . . . . . . . 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 17728	. . . . . . . . . . . . 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 7231	. . . . . . . . . . . 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 6540	. . . . . . . . . . 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 260	. . . . . . . . . 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 3112	. . . . . . . . 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 3112	. . . . . . . 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 7221	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(2nd ‘𝐸)𝑦) = (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
101		oveq2 7221	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
102		fveq2 6717	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩))
103		df-ov 7216	. . . . . . . . . . . . . 14 ⊢ (𝑔(1st ‘𝐸)𝑣) = ((1st ‘𝐸)‘⟨𝑔, 𝑣⟩)
104	102, 103	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st ‘𝐸)‘𝑦) = (𝑔(1st ‘𝐸)𝑣))
105	104	oveq2d 7229	. . . . . . . . . . . 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 6534	. . . . . . . . . . 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 5710	. . . . . . . . . 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 7220	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑔, 𝑣⟩))
109		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
110		fveq2 6717	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩))
111		df-ov 7216	. . . . . . . . . . . . . 14 ⊢ (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝐸)‘⟨𝑓, 𝑢⟩)
112	110, 111	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st ‘𝐸)‘𝑥) = (𝑓(1st ‘𝐸)𝑢))
113	112	oveq1d 7228	. . . . . . . . . . . 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 6534	. . . . . . . . . . 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 3120	. . . . . . . . . 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	syl5bb 286	. . . . . . . . 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 5710	. . . . . . . 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 237	. . . . . . 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 3130	. . . . . 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 3130	. . . . 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 470	. . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑄 ∈ Cat)
123	2	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
124		eqid 2737	. . . . . . . . . . 11 ⊢ (Id‘𝑄) = (Id‘𝑄)
125		eqid 2737	. . . . . . . . . . 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 17696	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩) = ⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
129	128	fveq2d 6721	. . . . . . . . 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 7216	. . . . . . . . 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 2796	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
133		eqid 2737	. . . . . . . . 9 ⊢ (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩)
134	20, 91, 124, 122, 126	catidcl 17185	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) ∈ (𝑓(𝐶 Nat 𝐷)𝑓))
135	4, 5, 125, 123, 127	catidcl 17185	. . . . . . . . 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 17727	. . . . . . . 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 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
138	4, 22, 137	funcf1 17372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
139	138, 127	ffvelrnd 6905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((1st ‘𝑓)‘𝑢) ∈ (Base‘𝐷))
140	22, 24, 26, 132, 139	catidcl 17185	. . . . . . . . . 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 17186	. . . . . . . . 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 17480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) = ((Id‘𝐷) ∘ (1st ‘𝑓)))
143	142	fveq1d 6719	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢))
144		fvco3 6810	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝐷) ∘ (1st ‘𝑓))‘𝑢) = ((Id‘𝐷)‘((1st ‘𝑓)‘𝑢)))
146	143, 145	eqtrd 2777	. . . . . . . . . 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 17376	. . . . . . . . . 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 7231	. . . . . . . . 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 17728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (𝑓(1st ‘𝐸)𝑢) = ((1st ‘𝑓)‘𝑢))
150	149	fveq2d 6721	. . . . . . . . 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 2787	. . . . . . . 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 2781	. . . . . . 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 3112	. . . . . 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 7231	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd ‘𝐸)𝑥) = (⟨𝑓, 𝑢⟩(2nd ‘𝐸)⟨𝑓, 𝑢⟩))
156		fveq2 6717	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘(𝑄 ×c 𝐶))‘𝑥) = ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩))
157	155, 156	fveq12d 6724	. . . . . . . 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 6721	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘𝐷)‘((1st ‘𝐸)‘𝑥)) = ((Id‘𝐷)‘(𝑓(1st ‘𝐸)𝑢)))
159	157, 158	eqeq12d 2753	. . . . . . 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 5710	. . . . . 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 237	. . . . 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 3130	. . . 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 1135	. . . . . 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 1135	. . . . . 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 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))) → 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
166		1st2nd2 7800	. . . . . . . . 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 2839	. . . . . . 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 5587	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑥) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑥) ∈ (Base‘𝐶)))
170	168, 169	sylib 221	. . . . . 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 1209	. . . . . . . . 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 7800	. . . . . . . . 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 2839	. . . . . . 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 5587	. . . . . . 7 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑦) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑦) ∈ (Base‘𝐶)))
176	174, 175	sylib 221	. . . . . 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 1210	. . . . . . . . 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 7800	. . . . . . . . 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 2839	. . . . . . 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 5587	. . . . . . 7 ⊢ (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st ‘𝑧) ∈ (𝐶 Func 𝐷) ∧ (2nd ‘𝑧) ∈ (Base‘𝐶)))
182	180, 181	sylib 221	. . . . . 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 1203	. . . . . . . . . 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 17687	. . . . . . . . . 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 2840	. . . . . . . . 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 7800	. . . . . . . . 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 2839	. . . . . . 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 5587	. . . . . . 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 221	. . . . . 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 1204	. . . . . . . . . 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 17687	. . . . . . . . . 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 2840	. . . . . . . . 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 7800	. . . . . . . . 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 2839	. . . . . . 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 5587	. . . . . . 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 221	. . . . . 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 17729	. . . . 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 7231	. . . . . 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 4792	. . . . . . . 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 7231	. . . . . . 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 7234	. . . . . 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 6724	. . . . 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 6721	. . . . . . . 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 6721	. . . . . . . 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 4792	. . . . . . 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 6721	. . . . . . 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 7231	. . . . . 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 7231	. . . . . . 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 6724	. . . . . 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 7231	. . . . . . 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 6724	. . . . . 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 7234	. . . . 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 2787	. . . 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 17371	. . 3 ⊢ (𝜑 → (1st ‘𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd ‘𝐸))
217		df-br 5054	. . 3 ⊢ ((1st ‘𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd ‘𝐸) ↔ ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
218	216, 217	sylib 221	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐸), (2nd ‘𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
219	17, 218	eqeltrd 2838	1 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408  ⦋csb 3811  ⟨cop 4547   class class class wbr 5053   × cxp 5549   ∘ ccom 5555  Rel wrel 5556   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  1^st c1st 7759  2^nd c2nd 7760  Basecbs 16760  Hom chom 16813  compcco 16814  Catccat 17167  Idccid 17168   Func cfunc 17360   Nat cnat 17448   FuncCat cfuc 17449   ×_c cxpc 17675   eval_F cevlf 17717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-struct 16700  df-slot 16735  df-ndx 16745  df-base 16761  df-hom 16826  df-cco 16827  df-cat 17171  df-cid 17172  df-func 17364  df-nat 17450  df-fuc 17451  df-xpc 17679  df-evlf 17721

This theorem is referenced by:  uncfcl  17743  uncf1  17744  uncf2  17745  yonedalem1  17780