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Theorem 1st2ndprf 17047
 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 (𝜑𝐹 ∈ (𝐶 Func 𝑇))
1st2ndprf.d (𝜑𝐷 ∈ Cat)
1st2ndprf.e (𝜑𝐸 ∈ Cat)
Assertion
Ref Expression
1st2ndprf (𝜑𝐹 = (((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.
StepHypRef Expression
1 eqid 2760 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 1st2ndprf.t . . . . . . 7 𝑇 = (𝐷 ×c 𝐸)
3 eqid 2760 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
4 eqid 2760 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
52, 3, 4xpcbas 17019 . . . . . 6 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6 relfunc 16723 . . . . . . 7 Rel (𝐶 Func 𝑇)
7 1st2ndprf.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝑇))
8 1st2ndbr 7384 . . . . . . 7 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
96, 7, 8sylancr 698 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
101, 5, 9funcf1 16727 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
1110feqmptd 6411 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
1210ffvelrnda 6522 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13 1st2nd2 7372 . . . . . . 7 (((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)) → ((1st𝐹)‘𝑥) = ⟨(1st ‘((1st𝐹)‘𝑥)), (2nd ‘((1st𝐹)‘𝑥))⟩)
1412, 13syl 17 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) = ⟨(1st ‘((1st𝐹)‘𝑥)), (2nd ‘((1st𝐹)‘𝑥))⟩)
157adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝑇))
16 1st2ndprf.d . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
17 1st2ndprf.e . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
18 eqid 2760 . . . . . . . . . . 11 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
192, 16, 17, 181stfcl 17038 . . . . . . . . . 10 (𝜑 → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
2019adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
21 simpr 479 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
221, 15, 20, 21cofu1 16745 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)))
23 eqid 2760 . . . . . . . . 9 (Hom ‘𝑇) = (Hom ‘𝑇)
2416adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2517adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
262, 5, 23, 24, 25, 18, 121stf1 17033 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)) = (1st ‘((1st𝐹)‘𝑥)))
2722, 26eqtrd 2794 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st𝐹)‘𝑥)))
28 eqid 2760 . . . . . . . . . . 11 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
292, 16, 17, 282ndfcl 17039 . . . . . . . . . 10 (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
3029adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
311, 15, 30, 21cofu1 16745 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)))
322, 5, 23, 24, 25, 28, 122ndf1 17036 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)) = (2nd ‘((1st𝐹)‘𝑥)))
3331, 32eqtrd 2794 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st𝐹)‘𝑥)))
3427, 33opeq12d 4561 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩ = ⟨(1st ‘((1st𝐹)‘𝑥)), (2nd ‘((1st𝐹)‘𝑥))⟩)
3514, 34eqtr4d 2797 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) = ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩)
3635mpteq2dva 4896 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
3711, 36eqtrd 2794 . . 3 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
381, 9funcfn2 16730 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39 fnov 6933 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4038, 39sylib 208 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
41 eqid 2760 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
429adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
43 simprl 811 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
44 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
451, 41, 23, 42, 43, 44funcf2 16729 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))
4645feqmptd 6411 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
472, 23relxpchom 17022 . . . . . . . . . 10 Rel (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))
4845ffvelrnda 6522 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))
49 1st2nd 7381 . . . . . . . . . 10 ((Rel (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)) ∧ ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) = ⟨(1st ‘((𝑥(2nd𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓))⟩)
5047, 48, 49sylancr 698 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘𝑓) = ⟨(1st ‘((𝑥(2nd𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓))⟩)
517ad2antrr 764 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝑇))
5219ad2antrr 764 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
5343adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
5444adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
55 simpr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
561, 51, 52, 53, 54, 41, 55cofu2 16747 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = ((((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦))‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
5716adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
5817adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐸 ∈ Cat)
5912adantrr 755 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6010ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrl 754 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
622, 5, 23, 57, 58, 18, 59, 611stf2 17034 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦)) = (1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
6362adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦)) = (1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
6463fveq1d 6354 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = ((1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
65 fvres 6368 . . . . . . . . . . . 12 (((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)) → ((1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
6648, 65syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
6756, 64, 663eqtrd 2798 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = (1st ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
6829ad2antrr 764 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
691, 51, 68, 53, 54, 41, 55cofu2 16747 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = ((((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦))‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
702, 5, 23, 57, 58, 28, 59, 612ndf2 17037 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦)) = (2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
7170adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦)) = (2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
7271fveq1d 6354 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = ((2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
73 fvres 6368 . . . . . . . . . . . 12 (((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)) → ((2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
7448, 73syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))‘((𝑥(2nd𝐹)𝑦)‘𝑓)) = (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
7569, 72, 743eqtrd 2798 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓) = (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓)))
7667, 75opeq12d 4561 . . . . . . . . 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(2nd𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓))⟩)
7750, 76eqtr4d 2797 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘𝑓) = ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩)
7877mpteq2dva 4896 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)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)⟩))
7946, 78eqtrd 2794 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))
80793impb 1108 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘((𝐷 1stF 𝐸) ∘func 𝐹))𝑦)‘𝑓), ((𝑥(2nd ‘((𝐷 2ndF 𝐸) ∘func 𝐹))𝑦)‘𝑓)⟩))
8180mpt2eq3dva 6884 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)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)⟩)))
8240, 81eqtrd 2794 . . 3 (𝜑 → (2nd𝐹) = (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)⟩)))
8337, 82opeq12d 4561 . 2 (𝜑 → ⟨(1st𝐹), (2nd𝐹)⟩ = ⟨(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)⟩))⟩)
84 1st2nd 7381 . . 3 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
856, 7, 84sylancr 698 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
86 eqid 2760 . . 3 (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)) = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹))
877, 19cofucl 16749 . . 3 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
887, 29cofucl 16749 . . 3 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
8986, 1, 41, 87, 88prfval 17040 . 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)⟩))⟩)
9083, 85, 893eqtr4d 2804 1 (𝜑𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ⟨cop 4327   class class class wbr 4804   ↦ cmpt 4881   × cxp 5264   ↾ cres 5268  Rel wrel 5271   Fn wfn 6044  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  1st c1st 7331  2nd c2nd 7332  Basecbs 16059  Hom chom 16154  Catccat 16526   Func cfunc 16715   ∘func ccofu 16717   ×c cxpc 17009   1stF c1stf 17010   2ndF c2ndf 17011   ⟨,⟩F cprf 17012 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-hom 16168  df-cco 16169  df-cat 16530  df-cid 16531  df-func 16719  df-cofu 16721  df-xpc 17013  df-1stf 17014  df-2ndf 17015  df-prf 17016 This theorem is referenced by: (None)
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