Theorem 1st2ndprf 18094

Description: Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
1st2ndprf.t	⊢ 𝑇 = (𝐷 ×c 𝐸)
1st2ndprf.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇))
1st2ndprf.d	⊢ (𝜑 → 𝐷 ∈ Cat)
1st2ndprf.e	⊢ (𝜑 → 𝐸 ∈ Cat)

Assertion

Ref	Expression
1st2ndprf	⊢ (𝜑 → 𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))

Proof of Theorem 1st2ndprf

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		1st2ndprf.t	. . . . . . 7 ⊢ 𝑇 = (𝐷 ×c 𝐸)
3		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
5	2, 3, 4	xpcbas 18066	. . . . . 6 ⊢ ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6		relfunc 17748	. . . . . . 7 ⊢ Rel (𝐶 Func 𝑇)
7		1st2ndprf.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇))
8		1st2ndbr 7974	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → (1st ‘𝐹)(𝐶 Func 𝑇)(2nd ‘𝐹))
9	6, 7, 8	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝑇)(2nd ‘𝐹))
10	1, 5, 9	funcf1 17752	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
11	10	feqmptd 6910	. . . 4 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
12	10	ffvelcdmda 7035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13		1st2nd2 7960	. . . . . . 7 ⊢ (((1st ‘𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) = ⟨(1st ‘((1st ‘𝐹)‘𝑥)), (2nd ‘((1st ‘𝐹)‘𝑥))⟩)
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ⟨(1st ‘((1st ‘𝐹)‘𝑥)), (2nd ‘((1st ‘𝐹)‘𝑥))⟩)
15	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝑇))
16		1st2ndprf.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
17		1st2ndprf.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
19	2, 16, 17, 18	1stfcl 18085	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
20	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
21		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
22	1, 15, 20, 21	cofu1 17770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)))
23		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
24	16	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
25	17	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
26	2, 5, 23, 24, 25, 18, 12	1stf1 18080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (1st ‘((1st ‘𝐹)‘𝑥)))
27	22, 26	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st ‘𝐹)‘𝑥)))
28		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
29	2, 16, 17, 28	2ndfcl 18086	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
30	29	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
31	1, 15, 30, 21	cofu1 17770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 2ndF 𝐸))‘((1st ‘𝐹)‘𝑥)))
32	2, 5, 23, 24, 25, 28, 12	2ndf1 18083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (2nd ‘((1st ‘𝐹)‘𝑥)))
33	31, 32	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st ‘𝐹)‘𝑥)))
34	27, 33	opeq12d 4838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩ = ⟨(1st ‘((1st ‘𝐹)‘𝑥)), (2nd ‘((1st ‘𝐹)‘𝑥))⟩)
35	14, 34	eqtr4d 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩)
36	35	mpteq2dva 5205	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
37	11, 36	eqtrd 2776	. . 3 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
38	1, 9	funcfn2 17755	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39		fnov 7487	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
40	38, 39	sylib 217	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
41		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝑇)(2nd ‘𝐹))
43		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
44		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
45	1, 41, 23, 42, 43, 44	funcf2 17754	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))
46	45	feqmptd 6910	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
47	2, 23	relxpchom 18069	. . . . . . . . . 10 ⊢ Rel (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))
48	45	ffvelcdmda 7035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))
49		1st2nd 7971	. . . . . . . . . 10 ⊢ ((Rel (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)) ∧ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) = ⟨(1st ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))⟩)
50	47, 48, 49	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) = ⟨(1st ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))⟩)
51	7	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝑇))
52	19	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
53	43	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
54	44	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
55		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
56	1, 51, 52, 53, 54, 41, 55	cofu2 17772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = ((((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
57	16	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
58	17	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐸 ∈ Cat)
59	12	adantrr 715	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
60	10	ffvelcdmda 7035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
61	60	adantrl 714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
62	2, 5, 23, 57, 58, 18, 59, 61	1stf2 18081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝐹)‘𝑦)) = (1st ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))))
63	62	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝐹)‘𝑦)) = (1st ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))))
64	63	fveq1d 6844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = ((1st ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
65	48	fvresd 6862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
66	56, 64, 65	3eqtrd 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = (1st ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
67	29	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
68	1, 51, 67, 53, 54, 41, 55	cofu2 17772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = ((((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
69	2, 5, 23, 57, 58, 28, 59, 61	2ndf2 18084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st ‘𝐹)‘𝑦)) = (2nd ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))))
70	69	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st ‘𝐹)‘𝑦)) = (2nd ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))))
71	70	fveq1d 6844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = ((2nd ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
72	48	fvresd 6862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((2nd ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦)))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (2nd ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
73	68, 71, 72	3eqtrd 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = (2nd ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
74	66, 73	opeq12d 4838	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩ = ⟨(1st ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓))⟩)
75	50, 74	eqtr4d 2779	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) = ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩)
76	75	mpteq2dva 5205	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))
77	46, 76	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))
78	77	3impb 1115	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))
79	78	mpoeq3dva 7434	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩)))
80	40, 79	eqtrd 2776	. . 3 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩)))
81	37, 80	opeq12d 4838	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))⟩)
82		1st2nd 7971	. . 3 ⊢ ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
83	6, 7, 82	sylancr 587	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
84		eqid 2736	. . 3 ⊢ (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)) = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹))
85	7, 19	cofucl 17774	. . 3 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
86	7, 29	cofucl 17774	. . 3 ⊢ (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
87	84, 1, 41, 85, 86	prfval 18087	. 2 ⊢ (𝜑 → (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)) = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))⟩)
88	81, 83, 87	3eqtr4d 2786	1 ⊢ (𝜑 → 𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631   ↾ cres 5635  Rel wrel 5638   Fn wfn 6491  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17083  Hom chom 17144  Catccat 17544   Func cfunc 17740   ∘_func ccofu 17742   ×_c cxpc 18056   1^st_F c1stf 18057   2^nd_F c2ndf 18058   ⟨,⟩_F cprf 18059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-func 17744  df-cofu 17746  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063

This theorem is referenced by: (None)