Theorem precofvalALT 49613

Description: Alternate proof of precofval 49612. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
precofval.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
precofval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
precofval.o	⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))
precofval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
precofval.e	⊢ (𝜑 → 𝐸 ∈ Cat)
precofval.k	⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))

Assertion

Ref	Expression
precofvalALT	⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)

Distinct variable groups:   𝐶,𝑎,𝑔,ℎ,𝑥   𝐷,𝑎,𝑔,ℎ,𝑥   𝐸,𝑎,𝑔,ℎ,𝑥   𝐹,𝑎,𝑔,ℎ,𝑥   𝑄,𝑎,𝑔,ℎ   𝑅,𝑎,𝑔,ℎ   𝜑,𝑎,𝑔,ℎ,𝑥

Allowed substitution hints:   𝑄(𝑥)   𝑅(𝑥)   𝐾(𝑥,𝑔,ℎ,𝑎)   ⚬ (𝑥,𝑔,ℎ,𝑎)

Proof of Theorem precofvalALT

Step	Hyp	Ref	Expression
1		precofval.o	. . 3 ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))
2		precofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
3	2	fucbas 17887	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4		relfunc 17786	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
5		precofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		1st2ndbr 7986	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
7	4, 5, 6	sylancr 587	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
8	7	funcrcl2 49324	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
9	7	funcrcl3 49325	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	2, 8, 9	fuccat 17897	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
11		precofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
12		precofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	11, 9, 12	fuccat 17897	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
14	11, 2	oveq12i 7370	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
15		eqid 2736	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
16	14, 15, 8, 9, 12	fucofunca 49605	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
17		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
18	11	fucbas 17887	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
19		eqid 2736	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
20	11, 19	fuchom 17888	. . 3 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑅)
21		eqid 2736	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
22	1, 3, 10, 13, 16, 5, 17, 18, 20, 21	tposcurf1 49544	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))))⟩)
23		df-ov 7361	. . . . 5 ⊢ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩)
24		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
25	8, 9, 12, 24	fucoelvv 49565	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
26		1st2nd2 7972	. . . . . . . . 9 ⊢ ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
29	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
30		relfunc 17786	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
31		1st2ndbr 7986	. . . . . . . . 9 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
32	30, 31	mpan 690	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 Func 𝐸) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
33	32	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
34		1st2nd 7983	. . . . . . . . . 10 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
35	30, 34	mpan 690	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐷 Func 𝐸) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
36	35	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
37		1st2nd 7983	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
38	4, 5, 37	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40	36, 39	opeq12d 4837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ⟨𝑔, 𝐹⟩ = ⟨⟨(1st ‘𝑔), (2nd ‘𝑔)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
41	28, 29, 33, 40	fuco11 49571	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
42	36, 39	oveq12d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔 ∘func 𝐹) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
43	41, 42	eqtr4d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩) = (𝑔 ∘func 𝐹))
44	23, 43	eqtrid 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = (𝑔 ∘func 𝐹))
45	44	mpteq2dva 5191	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)) = (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)))
46		eqid 2736	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
47	2, 21, 46, 5	fucid 17898	. . . . . . . . 9 ⊢ (𝜑 → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st ‘𝐹)))
48	47	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st ‘𝐹)))
49	48	oveq2d 7374	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹)) = (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝐷) ∘ (1st ‘𝐹))))
50	27	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
51		eqidd 2737	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨𝑔, 𝐹⟩ = ⟨𝑔, 𝐹⟩)
52		eqidd 2737	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨ℎ, 𝐹⟩ = ⟨ℎ, 𝐹⟩)
53		eqid 2736	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
54	2, 53, 46, 5	fucidcl 17892	. . . . . . . . 9 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
55	54	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ((Id‘𝐷) ∘ (1st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
56		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ))
57	50, 51, 52, 55, 56	fuco22a 49595	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝐷) ∘ (1st ‘𝐹))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥)))))
58		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
59		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
60		eqid 2736	. . . . . . . . . . . 12 ⊢ (Id‘𝐸) = (Id‘𝐸)
61	7	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
62	32	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
63	62	ad3antlr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
64		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
65	58, 59, 46, 60, 61, 63, 64	precofvallem 49611	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥)) = ((Id‘𝐸)‘((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))) ∧ ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸)))
66	65	simpld 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥)) = ((Id‘𝐸)‘((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))))
67	66	oveq2d 7374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥))) = ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((Id‘𝐸)‘((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)))))
68		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
69	12	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
70	65	simprd 495	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
71		eqid 2736	. . . . . . . . . 10 ⊢ (comp‘𝐸) = (comp‘𝐸)
72		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
73		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)))
74	73	simprd 495	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ℎ ∈ (𝐷 Func 𝐸))
75		1st2ndbr 7986	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)) → (1st ‘ℎ)(𝐷 Func 𝐸)(2nd ‘ℎ))
76	30, 74, 75	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘ℎ)(𝐷 Func 𝐸)(2nd ‘ℎ))
77	72, 59, 76	funcf1 17790	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘ℎ):(Base‘𝐷)⟶(Base‘𝐸))
78	7	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
79	58, 72, 78	funcf1 17790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
80	79	ffvelcdmda 7029	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
81	77, 80	ffvelcdmd 7030	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
82	56	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ))
83	19, 82	nat1st2nd 17878	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 ∈ (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(𝐷 Nat 𝐸)⟨(1st ‘ℎ), (2nd ‘ℎ)⟩))
84	19, 83, 72, 68, 80	natcl 17880	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘((1st ‘𝐹)‘𝑥)) ∈ (((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥))))
85	59, 68, 60, 69, 70, 71, 81, 84	catrid 17607	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((Id‘𝐸)‘((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)))) = (𝑎‘((1st ‘𝐹)‘𝑥)))
86	67, 85	eqtrd 2771	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥))) = (𝑎‘((1st ‘𝐹)‘𝑥)))
87	86	mpteq2dva 5191	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑎‘((1st ‘𝐹)‘𝑥))(⟨((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝑔)‘((1st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1st ‘ℎ)‘((1st ‘𝐹)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝑔)((1st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1st ‘𝐹))‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))
88	49, 57, 87	3eqtrd 2775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))
89	88	mpteq2dva 5191	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) → (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))) = (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))
90	89	3impb 1114	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)) → (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))) = (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))
91	90	mpoeq3dva 7435	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹)))) = (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))))
92	45, 91	opeq12d 4837	. 2 ⊢ (𝜑 → ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))))⟩ = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)
93	22, 92	eqtrd 2771	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622   ∘ ccom 5628  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Idccid 17588   Func cfunc 17778   ∘_func ccofu 17780   Nat cnat 17868   FuncCat cfuc 17869   ×_c cxpc 18091   curry_F ccurf 18133   swap_F cswapf 49504   ∘_F cfuco 49561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-cofu 17784  df-nat 17870  df-fuc 17871  df-xpc 18095  df-curf 18137  df-swapf 49505  df-fuco 49562

This theorem is referenced by: (None)