Theorem 1st2ndprf 18154

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 2732	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		1st2ndprf.t	. . . . . . 7 ⊢ 𝑇 = (𝐷 ×c 𝐸)
3		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
5	2, 3, 4	xpcbas 18126	. . . . . 6 ⊢ ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6		relfunc 17808	. . . . . . 7 ⊢ Rel (𝐶 Func 𝑇)
7		1st2ndprf.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇))
8		1st2ndbr 8024	. . . . . . 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 17812	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
11	10	feqmptd 6957	. . . 4 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
12	10	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13		1st2nd2 8010	. . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
19	2, 16, 17, 18	1stfcl 18145	. . . . . . . . . 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 17830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)))
23		eqid 2732	. . . . . . . . 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 18140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (1st ‘((1st ‘𝐹)‘𝑥)))
27	22, 26	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st ‘𝐹)‘𝑥)))
28		eqid 2732	. . . . . . . . . . 11 ⊢ (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
29	2, 16, 17, 28	2ndfcl 18146	. . . . . . . . . 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 17830	. . . . . . . 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 18143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st ‘𝐹)‘𝑥)) = (2nd ‘((1st ‘𝐹)‘𝑥)))
33	31, 32	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st ‘𝐹)‘𝑥)))
34	27, 33	opeq12d 4880	. . . . . 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 2775	. . . . 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 5247	. . . 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 2772	. . 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 17815	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39		fnov 7536	. . . . 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 2732	. . . . . . . . 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 17814	. . . . . . . 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 6957	. . . . . . 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 18129	. . . . . . . . . 10 ⊢ Rel (((1st ‘𝐹)‘𝑥)(Hom ‘𝑇)((1st ‘𝐹)‘𝑦))
48	45	ffvelcdmda 7083	. . . . . . . . . 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 8021	. . . . . . . . . 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 17832	. . . . . . . . . . 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 7083	. . . . . . . . . . . . . . 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 18141	. . . . . . . . . . . . 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 6890	. . . . . . . . . . 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 6908	. . . . . . . . . . 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 2776	. . . . . . . . . 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 17832	. . . . . . . . . . 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 18144	. . . . . . . . . . . . 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 6890	. . . . . . . . . . 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 6908	. . . . . . . . . . 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 2776	. . . . . . . . . 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 4880	. . . . . . . . 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 2775	. . . . . . . 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 5247	. . . . . . 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 2772	. . . . . 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 7482	. . . 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 2772	. . 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 4880	. 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 8021	. . 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 2732	. . 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 17834	. . 3 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
86	7, 29	cofucl 17834	. . 3 ⊢ (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
87	84, 1, 41, 85, 86	prfval 18147	. 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 2782	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 4633   class class class wbr 5147   ↦ cmpt 5230   × cxp 5673   ↾ cres 5677  Rel wrel 5680   Fn wfn 6535  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17140  Hom chom 17204  Catccat 17604   Func cfunc 17800   ∘_func ccofu 17802   ×_c cxpc 18116   1^st_F c1stf 18117   2^nd_F c2ndf 18118   ⟨,⟩_F cprf 18119

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17141  df-hom 17217  df-cco 17218  df-cat 17608  df-cid 17609  df-func 17804  df-cofu 17806  df-xpc 18120  df-1stf 18121  df-2ndf 18122  df-prf 18123

This theorem is referenced by: (None)