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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 (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 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)
52, 3, 4xpcbas 18066 . . . . . 6 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6 relfunc 17748 . . . . . . 7 Rel (𝐶 Func 𝑇)
7 1st2ndprf.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝑇))
8 1st2ndbr 7974 . . . . . . 7 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
96, 7, 8sylancr 587 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
101, 5, 9funcf1 17752 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
1110feqmptd 6910 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
1210ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13 1st2nd2 7960 . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝑇))
16 1st2ndprf.d . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
17 1st2ndprf.e . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
18 eqid 2736 . . . . . . . . . . 11 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
192, 16, 17, 181stfcl 18085 . . . . . . . . . 10 (𝜑 → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
2019adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
21 simpr 485 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
221, 15, 20, 21cofu1 17770 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)))
23 eqid 2736 . . . . . . . . 9 (Hom ‘𝑇) = (Hom ‘𝑇)
2416adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2517adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
262, 5, 23, 24, 25, 18, 121stf1 18080 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)) = (1st ‘((1st𝐹)‘𝑥)))
2722, 26eqtrd 2776 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st𝐹)‘𝑥)))
28 eqid 2736 . . . . . . . . . . 11 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
292, 16, 17, 282ndfcl 18086 . . . . . . . . . 10 (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
3029adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
311, 15, 30, 21cofu1 17770 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)))
322, 5, 23, 24, 25, 28, 122ndf1 18083 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)) = (2nd ‘((1st𝐹)‘𝑥)))
3331, 32eqtrd 2776 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st𝐹)‘𝑥)))
3427, 33opeq12d 4838 . . . . . 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 2779 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) = ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩)
3635mpteq2dva 5205 . . . 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 2776 . . 3 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
381, 9funcfn2 17755 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39 fnov 7487 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4038, 39sylib 217 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
41 eqid 2736 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
429adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
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))
451, 41, 23, 42, 43, 44funcf2 17754 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))
4645feqmptd 6910 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
472, 23relxpchom 18069 . . . . . . . . . 10 Rel (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))
4845ffvelcdmda 7035 . . . . . . . . . 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 7971 . . . . . . . . . 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 587 . . . . . . . . 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 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝑇))
5219ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
5343adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
5444adantr 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))
561, 51, 52, 53, 54, 41, 55cofu2 17772 . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
5817adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐸 ∈ Cat)
5912adantrr 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6010ffvelcdmda 7035 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrl 714 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
622, 5, 23, 57, 58, 18, 59, 611stf2 18081 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦)) = (1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
6362adantr 481 . . . . . . . . . . . 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 6844 . . . . . . . . . . 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)))
6548fvresd 6862 . . . . . . . . . . 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)))
6656, 64, 653eqtrd 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(2nd𝐹)𝑦)‘𝑓)))
6729ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
681, 51, 67, 53, 54, 41, 55cofu2 17772 . . . . . . . . . . 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)))
692, 5, 23, 57, 58, 28, 59, 612ndf2 18084 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦)) = (2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
7069adantr 481 . . . . . . . . . . . 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))))
7170fveq1d 6844 . . . . . . . . . . 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)))
7248fvresd 6862 . . . . . . . . . . 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)))
7368, 71, 723eqtrd 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(2nd𝐹)𝑦)‘𝑓)))
7466, 73opeq12d 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(2nd𝐹)𝑦)‘𝑓)), (2nd ‘((𝑥(2nd𝐹)𝑦)‘𝑓))⟩)
7550, 74eqtr4d 2779 . . . . . . . 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)⟩)
7675mpteq2dva 5205 . . . . . . 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)⟩))
7746, 76eqtrd 2776 . . . . . 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)⟩))
78773impb 1115 . . . . 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)⟩))
7978mpoeq3dva 7434 . . . 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)⟩)))
8040, 79eqtrd 2776 . . 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)⟩)))
8137, 80opeq12d 4838 . 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)⟩))⟩)
82 1st2nd 7971 . . 3 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
836, 7, 82sylancr 587 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
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))
857, 19cofucl 17774 . . 3 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
867, 29cofucl 17774 . . 3 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
8784, 1, 41, 85, 86prfval 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)⟩))⟩)
8881, 83, 873eqtr4d 2786 1 (𝜑𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))
Colors of variables: wff setvar class
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  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  Catccat 17544   Func cfunc 17740  func ccofu 17742   ×c cxpc 18056   1stF c1stf 18057   2ndF 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)
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