Theorem precofvalALT 49113

Description: Alternate proof of precofval 49112. (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 (𝑄swapF𝑅))))
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 (𝑄swapF𝑅))))
2		precofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
3	2	fucbas 17980	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4		relfunc 17879	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
5		precofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		1st2ndbr 8049	. . . . . 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 48937	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
9	7	funcrcl3 48938	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	2, 8, 9	fuccat 17990	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
11		precofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
12		precofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	11, 9, 12	fuccat 17990	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
14	11, 2	oveq12i 7425	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
15		eqid 2734	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
16	14, 15, 8, 9, 12	fucofunca 49105	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
17		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
18	11	fucbas 17980	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
19		eqid 2734	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
20	11, 19	fuchom 17981	. . 3 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑅)
21		eqid 2734	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
22	1, 3, 10, 13, 16, 5, 17, 18, 20, 21	tposcurf1 49044	. 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 7416	. . . . 5 ⊢ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩)
24		eqidd 2735	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
25	8, 9, 12, 24	fucoelvv 49065	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
26		1st2nd2 8035	. . . . . . . . 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 17879	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
31		1st2ndbr 8049	. . . . . . . . 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 8046	. . . . . . . . . 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 8046	. . . . . . . . . 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 4861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ⟨𝑔, 𝐹⟩ = ⟨⟨(1st ‘𝑔), (2nd ‘𝑔)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
41	28, 29, 33, 40	fuco11 49071	. . . . . 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 7431	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔 ∘func 𝐹) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
43	41, 42	eqtr4d 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝑔, 𝐹⟩) = (𝑔 ∘func 𝐹))
44	23, 43	eqtrid 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = (𝑔 ∘func 𝐹))
45	44	mpteq2dva 5222	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)) = (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)))
46		eqid 2734	. . . . . . . . . 10 ⊢ (Id‘𝐷) = (Id‘𝐷)
47	2, 21, 46, 5	fucid 17991	. . . . . . . . 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 7429	. . . . . . 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 2735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨𝑔, 𝐹⟩ = ⟨𝑔, 𝐹⟩)
52		eqidd 2735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔 ∈ (𝐷 Func 𝐸) ∧ ℎ ∈ (𝐷 Func 𝐸))) ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → ⟨ℎ, 𝐹⟩ = ⟨ℎ, 𝐹⟩)
53		eqid 2734	. . . . . . . . . 10 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
54	2, 53, 46, 5	fucidcl 17985	. . . . . . . . 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 49095	. . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
59		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
60		eqid 2734	. . . . . . . . . . . 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 49111	. . . . . . . . . . 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 7429	. . . . . . . . 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 2734	. . . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (comp‘𝐸) = (comp‘𝐸)
72		eqid 2734	. . . . . . . . . . . 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 8049	. . . . . . . . . . . . 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 17883	. . . . . . . . . . 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 17883	. . . . . . . . . . . 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 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 17971	. . . . . . . . . . 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 17973	. . . . . . . . . 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 17699	. . . . . . . . 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 2769	. . . . . . . 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 5222	. . . . . . 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 2773	. . . . . 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 5222	. . . . 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 7492	. . 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 4861	. 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 2769	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 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612   class class class wbr 5123   ↦ cmpt 5205   × cxp 5663   ∘ ccom 5669  Rel wrel 5670  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994  2^nd c2nd 7995  Basecbs 17230  Hom chom 17285  compcco 17286  Catccat 17679  Idccid 17680   Func cfunc 17871   ∘_func ccofu 17873   Nat cnat 17961   FuncCat cfuc 17962   ×_c cxpc 18184   curry_F ccurf 18226  swap_Fcswapf 49010   ∘_F cfuco 49061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-nat 17963  df-fuc 17964  df-xpc 18188  df-curf 18230  df-swapf 49011  df-fuco 49062

This theorem is referenced by: (None)