Theorem precofvalALT 50475

Description: Alternate proof of precofval 50474. (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 18138	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4		relfunc 18037	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
5		precofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		1st2ndbr 8053	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
7	4, 5, 6	sylancr 599	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
8	7	funcrcl2 50186	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
9	7	funcrcl3 50187	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	2, 8, 9	fuccat 18148	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
11		precofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
12		precofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	11, 9, 12	fuccat 18148	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
14	11, 2	oveq12i 7432	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
15		eqid 2761	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
16	14, 15, 8, 9, 12	fucofunca 50467	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
17		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
18	11	fucbas 18138	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
19		eqid 2761	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
20	11, 19	fuchom 18139	. . 3 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑅)
21		eqid 2761	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
22	1, 3, 10, 13, 16, 5, 17, 18, 20, 21	tposcurf1 50406	. 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 7423	. . . . 5 ⊢ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩)
24		eqidd 2762	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
25	8, 9, 12, 24	fucoelvv 50427	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
26		1st2nd2 8040	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
28	27	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
29	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
30		relfunc 18037	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
31		1st2ndbr 8053	. . . . . . . . 9 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
32	30, 31	mpan 703	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐷 Func 𝐸) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
33	32	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
34		1st2nd 8050	. . . . . . . . . 10 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
35	30, 34	mpan 703	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐷 Func 𝐸) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
36	35	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
37		1st2nd 8050	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
38	4, 5, 37	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40	36, 39	opeq12d 4841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ⟨𝑔, 𝐹⟩ = ⟨⟨(1st ‘𝑔), (2nd ‘𝑔)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
41	28, 29, 33, 40	fuco11 50433	. . . . . 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 7438	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔 ∘func 𝐹) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
43	41, 42	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩) = (𝑔 ∘func 𝐹))
44	23, 43	eqtrid 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = (𝑔 ∘func 𝐹))
45	44	mpteq2dva 5198	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)) = (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)))
46		eqid 2761	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
47	2, 21, 46, 5	fucid 18149	. . . . . . . . 9 ⊢ (𝜑 → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1st ‘𝐹)))
48	47	ad2antrr 739	. . . . . . . 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 7436	. . . . . . 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 739	. . . . . . . 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 2762	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨𝑔, 𝐹⟩ = ⟨𝑔, 𝐹⟩)
52		eqidd 2762	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨ℎ, 𝐹⟩ = ⟨ℎ, 𝐹⟩)
53		eqid 2761	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
54	2, 53, 46, 5	fucidcl 18143	. . . . . . . . 9 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
55	54	ad2antrr 739	. . . . . . . 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 490	. . . . . . . 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 50457	. . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
59		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
60		eqid 2761	. . . . . . . . . . . 12 ⊢ (Id‘𝐸) = (Id‘𝐸)
61	7	ad3antrrr 743	. . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)) → (1st ‘𝑔)(𝐷 Func 𝐸)(2nd ‘𝑔))
63	62	ad3antlr 744	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 50473	. . . . . . . . . . 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 500	. . . . . . . . . 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 7436	. . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
69	12	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
70	65	simprd 501	. . . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (comp‘𝐸) = (comp‘𝐸)
72		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
73		simpllr 788	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 8053	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸)) → (1st ‘ℎ)(𝐷 Func 𝐸)(2nd ‘ℎ))
76	30, 74, 75	sylancr 599	. . . . . . . . . . . 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 18041	. . . . . . . . . . 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 739	. . . . . . . . . . . . 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 18041	. . . . . . . . . . . 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 7084	. . . . . . . . . . 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 7085	. . . . . . . . . 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 486	. . . . . . . . . . . 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 18129	. . . . . . . . . . 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 18131	. . . . . . . . . 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 17858	. . . . . . . . 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 2796	. . . . . . . 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 5198	. . . . . . 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 2800	. . . . . 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 5198	. . . . 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 1132	. . . 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 7497	. . 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 4841	. 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 2796	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)))))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649   ∘ ccom 5655  Rel wrel 5656  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839   Func cfunc 18029   ∘_func ccofu 18031   Nat cnat 18119   FuncCat cfuc 18120   ×_c cxpc 18342   curry_F ccurf 18384   swap_F cswapf 50366   ∘_F cfuco 50423

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 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035  df-nat 18121  df-fuc 18122  df-xpc 18346  df-curf 18388  df-swapf 50367  df-fuco 50424

This theorem is used by: (None)