Theorem 1st2ndprf 17458

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 2823	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		1st2ndprf.t	. . . . . . 7 ⊢ 𝑇 = (𝐷 ×c 𝐸)
3		eqid 2823	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2823	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
5	2, 3, 4	xpcbas 17430	. . . . . 6 ⊢ ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6		relfunc 17134	. . . . . . 7 ⊢ Rel (𝐶 Func 𝑇)
7		1st2ndprf.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇))
8		1st2ndbr 7743	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → (1st ‘𝐹)(𝐶 Func 𝑇)(2nd ‘𝐹))
9	6, 7, 8	sylancr 589	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝑇)(2nd ‘𝐹))
10	1, 5, 9	funcf1 17138	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
11	10	feqmptd 6735	. . . 4 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
12	10	ffvelrnda 6853	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13		1st2nd2 7730	. . . . . . 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 483	. . . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
19	2, 16, 17, 18	1stfcl 17449	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
20	19	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
21		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
22	1, 15, 20, 21	cofu1 17156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)))
23		eqid 2823	. . . . . . . . 9 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
24	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
25	17	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
26	2, 5, 23, 24, 25, 18, 12	1stf1 17444	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (1st ‘((1st ‘𝐹)‘𝑥)))
27	22, 26	eqtrd 2858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st ‘𝐹)‘𝑥)))
28		eqid 2823	. . . . . . . . . . 11 ⊢ (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
29	2, 16, 17, 28	2ndfcl 17450	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
30	29	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
31	1, 15, 30, 21	cofu1 17156	. . . . . . . 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 17447	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (2nd ‘((1st ‘𝐹)‘𝑥)))
33	31, 32	eqtrd 2858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st ‘𝐹)‘𝑥)))
34	27, 33	opeq12d 4813	. . . . . 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 2861	. . . . 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 5163	. . . 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 2858	. . 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 17141	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39		fnov 7284	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
40	38, 39	sylib 220	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
41		eqid 2823	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42	9	adantr 483	. . . . . . . . 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 17140	. . . . . . . 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 6735	. . . . . . 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 17433	. . . . . . . . . 10 ⊢ Rel (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))
48	45	ffvelrnda 6853	. . . . . . . . . 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 7740	. . . . . . . . . 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 589	. . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
54	44	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
55		simpr 487	. . . . . . . . . . . 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 17158	. . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
58	17	adantr 483	. . . . . . . . . . . . . 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	ffvelrnda 6853	. . . . . . . . . . . . . . 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 17445	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 6674	. . . . . . . . . . 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 6692	. . . . . . . . . . 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 2862	. . . . . . . . . 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 17158	. . . . . . . . . . 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 17448	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 6674	. . . . . . . . . . 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 6692	. . . . . . . . . . 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 2862	. . . . . . . . . 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 4813	. . . . . . . . 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 2861	. . . . . . . 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 5163	. . . . . . 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 2858	. . . . . 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 1111	. . . . 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 7233	. . . 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 2858	. . 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 4813	. 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 7740	. . 3 ⊢ ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
83	6, 7, 82	sylancr 589	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
84		eqid 2823	. . 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 17160	. . 3 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
86	7, 29	cofucl 17160	. . 3 ⊢ (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
87	84, 1, 41, 85, 86	prfval 17451	. 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 2868	1 ⊢ (𝜑 → 𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555   ↾ cres 5559  Rel wrel 5562   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  1^st c1st 7689  2^nd c2nd 7690  Basecbs 16485  Hom chom 16578  Catccat 16937   Func cfunc 17126   ∘_func ccofu 17128   ×_c cxpc 17420   1^st_F c1stf 17421   2^nd_F c2ndf 17422   ⟨,⟩_F cprf 17423

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-hom 16591  df-cco 16592  df-cat 16941  df-cid 16942  df-func 17130  df-cofu 17132  df-xpc 17424  df-1stf 17425  df-2ndf 17426  df-prf 17427

This theorem is referenced by: (None)