Theorem curf1cl 18395

Description: The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a	⊢ 𝐴 = (Base‘𝐶)
curfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfval.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b	⊢ 𝐵 = (Base‘𝐷)
curf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf1.k	⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)

Assertion

Ref	Expression
curf1cl	⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))

Proof of Theorem curf1cl

Dummy variables 𝑔 𝑦 𝑧 ℎ 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfval.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		curfval.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
3		curfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		curfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
7		curf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
8		curf1.k	. . . 4 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
9		eqid 2761	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2761	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18392	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12	6	fvexi 6897	. . . . . . 7 ⊢ 𝐵 ∈ V
13	12	mptex 7227	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8090	. . . . . 6 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
15	13, 14	op1std 8009	. . . . 5 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)))
16	11, 15	syl 18	. . . 4 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)))
17	13, 14	op2ndd 8010	. . . . 5 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
18	11, 17	syl 18	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
19	16, 18	opeq12d 4841	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
20	11, 19	eqtr4d 2799	. 2 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
21		eqid 2761	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
22		eqid 2761	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
23		eqid 2761	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		eqid 2761	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
25		eqid 2761	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
26		eqid 2761	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
27		funcrcl 18031	. . . . . 6 ⊢ (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
28	5, 27	syl 18	. . . . 5 ⊢ (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
29	28	simprd 501	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
30		eqid 2761	. . . . . . . . 9 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
31	30, 2, 6	xpcbas 18345	. . . . . . . 8 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
32		relfunc 18030	. . . . . . . . 9 ⊢ Rel ((𝐶 ×c 𝐷) Func 𝐸)
33		1st2ndbr 8051	. . . . . . . . 9 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
34	32, 5, 33	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
35	31, 21, 34	funcf1 18034	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
37	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
38		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
39	36, 37, 38	fovcdmd 7591	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘𝐹)𝑦) ∈ (Base‘𝐸))
40	16, 39	fmpt3d 7114	. . . 4 ⊢ (𝜑 → (1st ‘𝐾):𝐵⟶(Base‘𝐸))
41		eqid 2761	. . . . . 6 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
42		ovex 7451	. . . . . . 7 ⊢ (𝑦(Hom ‘𝐷)𝑧) ∈ V
43	42	mptex 7227	. . . . . 6 ⊢ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
44	41, 43	fnmpoi 8079	. . . . 5 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)
45	18	fneq1d 6630	. . . . 5 ⊢ (𝜑 → ((2nd ‘𝐾) Fn (𝐵 × 𝐵) ↔ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)))
46	44, 45	mpbiri 261	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) Fn (𝐵 × 𝐵))
47	18	oveqd 7435	. . . . . 6 ⊢ (𝜑 → (𝑦(2nd ‘𝐾)𝑧) = (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧))
48	41	ovmpt4g 7565	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V) → (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
49	43, 48	mp3an3 1479	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
50	47, 49	sylan9eq 2816	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
51		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
52	34	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
53	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋 ∈ 𝐴)
54		simplrl 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ 𝐵)
55	53, 54	opelxpd 5690	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
56		simplrr 790	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ 𝐵)
57	53, 56	opelxpd 5690	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
58	31, 51, 22, 52, 55, 57	funcf2 18036	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩):(⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩)⟶(((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)))
59		eqid 2761	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
60	30, 31, 59, 9, 51, 55, 57	xpchom 18347	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩) = (((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩))))
61	3	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
62	4	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
63	5	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
64	1, 2, 61, 62, 63, 6, 53, 8, 54	curf11 18393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦))
65		df-ov 7421	. . . . . . . . . 10 ⊢ (𝑋(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)
66	64, 65	eqtr2di 2813	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐹)‘⟨𝑋, 𝑦⟩) = ((1st ‘𝐾)‘𝑦))
67	1, 2, 61, 62, 63, 6, 53, 8, 56	curf11 18393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐾)‘𝑧) = (𝑋(1st ‘𝐹)𝑧))
68		df-ov 7421	. . . . . . . . . 10 ⊢ (𝑋(1st ‘𝐹)𝑧) = ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)
69	67, 68	eqtr2di 2813	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩) = ((1st ‘𝐾)‘𝑧))
70	66, 69	oveq12d 7436	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)) = (((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧)))
71	60, 70	feq23d 6702	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩):(⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩)⟶(((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)) ↔ (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩):(((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))⟶(((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧))))
72	58, 71	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩):(((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))⟶(((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧)))
73	2, 59, 10, 61, 53	catidcl 17849	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
74		op1stg 8011	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
75	53, 54, 74	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
76		op1stg 8011	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
77	53, 56, 76	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
78	75, 77	oveq12d 7436	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) = (𝑋(Hom ‘𝐶)𝑋))
79	73, 78	eleqtrrd 2864	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)))
80		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
81		op2ndg 8012	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
82	53, 54, 81	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
83		op2ndg 8012	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
84	53, 56, 83	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
85	82, 84	oveq12d 7436	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)) = (𝑦(Hom ‘𝐷)𝑧))
86	80, 85	eleqtrrd 2864	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))
87	72, 79, 86	fovcdmd 7591	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ (((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧)))
88	50, 87	fmpt3d 7114	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧)))
89	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
90	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
91		eqid 2761	. . . . . . . . 9 ⊢ (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
92	30, 89, 90, 2, 6, 10, 23, 91, 37, 38	xpcid 18356	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
93	92	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩))
94		df-ov 7421	. . . . . . 7 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
95	93, 94	eqtr4di 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
96	34	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
97		opelxpi 5688	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
98	7, 97	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
99	31, 91, 24, 96, 98	funcid 18038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((Id‘𝐸)‘((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)))
100	95, 99	eqtr3d 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)))
101	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
102	6, 9, 23, 90, 38	catidcl 17849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦))
103	1, 2, 89, 90, 101, 6, 37, 8, 38, 9, 10, 38, 102	curf12 18394	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦(2nd ‘𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
104	1, 2, 89, 90, 101, 6, 37, 8, 38	curf11 18393	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦))
105	104, 65	eqtrdi 2812	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = ((1st ‘𝐹)‘⟨𝑋, 𝑦⟩))
106	105	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘𝐸)‘((1st ‘𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐹)‘⟨𝑋, 𝑦⟩)))
107	100, 103, 106	3eqtr4d 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦(2nd ‘𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐾)‘𝑦)))
108	7	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋 ∈ 𝐴)
109		simp21 1225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦 ∈ 𝐵)
110		simp22 1226	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧 ∈ 𝐵)
111		eqid 2761	. . . . . . . . . 10 ⊢ (comp‘𝐶) = (comp‘𝐶)
112		eqid 2761	. . . . . . . . . 10 ⊢ (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
113		simp23 1227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤 ∈ 𝐵)
114	3	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
115	2, 59, 10, 114, 108	catidcl 17849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
116		simp3l 1220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
117		simp3r 1221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))
118	30, 2, 6, 59, 9, 108, 109, 108, 110, 111, 25, 112, 108, 113, 115, 116, 115, 117	xpcco2 18354	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑋), ℎ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩) = ⟨(((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
119	2, 59, 10, 114, 108, 111, 108, 115	catlid 17850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)) = ((Id‘𝐶)‘𝑋))
120	119	opeq1d 4839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨(((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩ = ⟨((Id‘𝐶)‘𝑋), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
121	118, 120	eqtrd 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑋), ℎ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩) = ⟨((Id‘𝐶)‘𝑋), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
122	121	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ℎ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩))
123		df-ov 7421	. . . . . . 7 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
124	122, 123	eqtr4di 2814	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ℎ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)))
125	34	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
126	108, 109	opelxpd 5690	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
127	108, 110	opelxpd 5690	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
128	108, 113	opelxpd 5690	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
129	115, 116	opelxpd 5690	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
130	30, 2, 6, 59, 9, 108, 109, 108, 110, 51	xpchom2 18353	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
131	129, 130	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩))
132	115, 117	opelxpd 5690	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ℎ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
133	30, 2, 6, 59, 9, 108, 110, 108, 113, 51	xpchom2 18353	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
134	132, 133	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ℎ⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
135	31, 51, 112, 26, 125, 126, 127, 128, 131, 134	funcco 18039	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ℎ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = (((⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ℎ⟩)(⟨((1st ‘𝐹)‘⟨𝑋, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
136	124, 135	eqtr3d 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ℎ⟩)(⟨((1st ‘𝐹)‘⟨𝑋, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
137	4	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
138	5	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
139	6, 9, 25, 137, 109, 110, 113, 116, 117	catcocl 17852	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐷)𝑤))
140	1, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 113, 139	curf12 18394	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑤)‘(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)))
141	1, 2, 114, 137, 138, 6, 108, 8, 109	curf11 18393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦))
142	141, 65	eqtrdi 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑦) = ((1st ‘𝐹)‘⟨𝑋, 𝑦⟩))
143	1, 2, 114, 137, 138, 6, 108, 8, 110	curf11 18393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑧) = (𝑋(1st ‘𝐹)𝑧))
144	143, 68	eqtrdi 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑧) = ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩))
145	142, 144	opeq12d 4841	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)⟩ = ⟨((1st ‘𝐹)‘⟨𝑋, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)⟩)
146	1, 2, 114, 137, 138, 6, 108, 8, 113	curf11 18393	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑤) = (𝑋(1st ‘𝐹)𝑤))
147		df-ov 7421	. . . . . . . 8 ⊢ (𝑋(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑋, 𝑤⟩)
148	146, 147	eqtrdi 2812	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑤) = ((1st ‘𝐹)‘⟨𝑋, 𝑤⟩))
149	145, 148	oveq12d 7436	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)⟩(comp‘𝐸)((1st ‘𝐾)‘𝑤)) = (⟨((1st ‘𝐹)‘⟨𝑋, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑤⟩)))
150	1, 2, 114, 137, 138, 6, 108, 8, 110, 9, 10, 113, 117	curf12 18394	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘𝐾)𝑤)‘ℎ) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)ℎ))
151		df-ov 7421	. . . . . . 7 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)ℎ) = ((⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ℎ⟩)
152	150, 151	eqtrdi 2812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘𝐾)𝑤)‘ℎ) = ((⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ℎ⟩))
153	1, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 110, 116	curf12 18394	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))
154		df-ov 7421	. . . . . . 7 ⊢ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)
155	153, 154	eqtrdi 2812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = ((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩))
156	149, 152, 155	oveq123d 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2nd ‘𝐾)𝑤)‘ℎ)(⟨((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)⟩(comp‘𝐸)((1st ‘𝐾)‘𝑤))((𝑦(2nd ‘𝐾)𝑧)‘𝑔)) = (((⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ℎ⟩)(⟨((1st ‘𝐹)‘⟨𝑋, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
157	136, 140, 156	3eqtr4d 2806	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑤)‘(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((𝑧(2nd ‘𝐾)𝑤)‘ℎ)(⟨((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)⟩(comp‘𝐸)((1st ‘𝐾)‘𝑤))((𝑦(2nd ‘𝐾)𝑧)‘𝑔)))
158	6, 21, 9, 22, 23, 24, 25, 26, 4, 29, 40, 46, 88, 107, 157	isfuncd 18033	. . 3 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
159		df-br 5104	. . 3 ⊢ ((1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾) ↔ ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Func 𝐸))
160	158, 159	sylib 221	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Func 𝐸))
161	20, 160	eqeltrd 2861	1 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  Rel wrel 5656   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  Idccid 17832   Func cfunc 18022   ×_c cxpc 18335   curry_F ccurf 18377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-func 18026  df-xpc 18339  df-curf 18381

This theorem is used by:  curf2cl  18398  curfcl  18399  tposcurf1cl  50373  postcofcl  50442