Theorem precofvalALT 50639

Description: Alternate proof of precofval 50638. (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 18147	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4		relfunc 18046	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
5		precofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		1st2ndbr 8057	. . . . . 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 50329	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
9	7	funcrcl3 50330	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	2, 8, 9	fuccat 18157	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
11		precofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
12		precofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	11, 9, 12	fuccat 18157	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
14	11, 2	oveq12i 7436	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
15		eqid 2760	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
16	14, 15, 8, 9, 12	fucofunca 50618	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
17		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
18	11	fucbas 18147	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
19		eqid 2760	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
20	11, 19	fuchom 18148	. . 3 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑅)
21		eqid 2760	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
22	1, 3, 10, 13, 16, 5, 17, 18, 20, 21	tposcurf1 50557	. 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 7427	. . . . 5 ⊢ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩)
24		eqidd 2761	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
25	8, 9, 12, 24	fucoelvv 50578	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
26		1st2nd2 8044	. . . . . . . . 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 18046	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
31		1st2ndbr 8057	. . . . . . . . 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 8054	. . . . . . . . . 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 8054	. . . . . . . . . 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 4847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ⟨𝑔, 𝐹⟩ = ⟨⟨(1st ‘𝑔), (2nd ‘𝑔)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
41	28, 29, 33, 40	fuco11 50584	. . . . . 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 7442	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔 ∘func 𝐹) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
43	41, 42	eqtr4d 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩) = (𝑔 ∘func 𝐹))
44	23, 43	eqtrid 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = (𝑔 ∘func 𝐹))
45	44	mpteq2dva 5197	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)) = (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)))
46		eqid 2760	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
47	2, 21, 46, 5	fucid 18158	. . . . . . . . 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 7440	. . . . . . 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 2761	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨𝑔, 𝐹⟩ = ⟨𝑔, 𝐹⟩)
52		eqidd 2761	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨ℎ, 𝐹⟩ = ⟨ℎ, 𝐹⟩)
53		eqid 2760	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
54	2, 53, 46, 5	fucidcl 18152	. . . . . . . . 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 50608	. . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
59		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
60		eqid 2760	. . . . . . . . . . . 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 50637	. . . . . . . . . . 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 7440	. . . . . . . . 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 2760	. . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (comp‘𝐸) = (comp‘𝐸)
72		eqid 2760	. . . . . . . . . . . 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 8057	. . . . . . . . . . . . 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 18050	. . . . . . . . . . 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 18050	. . . . . . . . . . . 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 7089	. . . . . . . . . . 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 7090	. . . . . . . . . 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 18138	. . . . . . . . . . 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 18140	. . . . . . . . . 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 17867	. . . . . . . . 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 2795	. . . . . . . 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 5197	. . . . . . 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 2799	. . . . . 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 5197	. . . . 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 7501	. . 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 4847	. 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 2795	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 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   × cxp 5649   ∘ ccom 5655  Rel wrel 5656  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8003  2^nd c2nd 8004  Basecbs 17396  Hom chom 17448  compcco 17449  Catccat 17847  Idccid 17848   Func cfunc 18038   ∘_func ccofu 18040   Nat cnat 18128   FuncCat cfuc 18129   ×_c cxpc 18351   curry_F ccurf 18393   swap_F cswapf 50517   ∘_F cfuco 50574

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-map 8857  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-struct 17334  df-slot 17369  df-ndx 17381  df-base 17397  df-hom 17461  df-cco 17462  df-cat 17851  df-cid 17852  df-func 18042  df-cofu 18044  df-nat 18130  df-fuc 18131  df-xpc 18355  df-curf 18397  df-swapf 50518  df-fuco 50575

This theorem is used by: (None)