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Theorem 1st2ndprf 17444
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 2818 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 1st2ndprf.t . . . . . . 7 𝑇 = (𝐷 ×c 𝐸)
3 eqid 2818 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
4 eqid 2818 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
52, 3, 4xpcbas 17416 . . . . . 6 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6 relfunc 17120 . . . . . . 7 Rel (𝐶 Func 𝑇)
7 1st2ndprf.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝑇))
8 1st2ndbr 7730 . . . . . . 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 17124 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
1110feqmptd 6726 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
1210ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13 1st2nd2 7717 . . . . . . 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 2818 . . . . . . . . . . 11 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
192, 16, 17, 181stfcl 17435 . . . . . . . . . 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 17142 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)))
23 eqid 2818 . . . . . . . . 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 17430 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)) = (1st ‘((1st𝐹)‘𝑥)))
2722, 26eqtrd 2853 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st𝐹)‘𝑥)))
28 eqid 2818 . . . . . . . . . . 11 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
292, 16, 17, 282ndfcl 17436 . . . . . . . . . 10 (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
3029adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
311, 15, 30, 21cofu1 17142 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)))
322, 5, 23, 24, 25, 28, 122ndf1 17433 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)) = (2nd ‘((1st𝐹)‘𝑥)))
3331, 32eqtrd 2853 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st𝐹)‘𝑥)))
3427, 33opeq12d 4803 . . . . . 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 2856 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) = ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩)
3635mpteq2dva 5152 . . . 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 2853 . . 3 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
381, 9funcfn2 17127 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39 fnov 7271 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4038, 39sylib 219 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
41 eqid 2818 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
429adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
43 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
44 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
451, 41, 23, 42, 43, 44funcf2 17126 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))
4645feqmptd 6726 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
472, 23relxpchom 17419 . . . . . . . . . 10 Rel (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))
4845ffvelrnda 6843 . . . . . . . . . 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 7727 . . . . . . . . . 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 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝑇))
5219ad2antrr 722 . . . . . . . . . . . 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 17144 . . . . . . . . . . 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 713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6010ffvelrnda 6843 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrl 712 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
622, 5, 23, 57, 58, 18, 59, 611stf2 17431 . . . . . . . . . . . . 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 6665 . . . . . . . . . . 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 6683 . . . . . . . . . . 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 2857 . . . . . . . . . 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 722 . . . . . . . . . . . 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 17144 . . . . . . . . . . 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 17434 . . . . . . . . . . . . 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 6665 . . . . . . . . . . 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 6683 . . . . . . . . . . 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 2857 . . . . . . . . . 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 4803 . . . . . . . . 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 2856 . . . . . . . 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 5152 . . . . . . 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 2853 . . . . . 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 1107 . . . . 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 7220 . . . 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 2853 . . 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 4803 . 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 7727 . . 3 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
836, 7, 82sylancr 587 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
84 eqid 2818 . . 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 17146 . . 3 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
867, 29cofucl 17146 . . 3 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
8784, 1, 41, 85, 86prfval 17437 . 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 2863 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 1528  wcel 2105  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  cres 5550  Rel wrel 5553   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  Catccat 16923   Func cfunc 17112  func ccofu 17114   ×c cxpc 17406   1stF c1stf 17407   2ndF c2ndf 17408   ⟨,⟩F cprf 17409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-func 17116  df-cofu 17118  df-xpc 17410  df-1stf 17411  df-2ndf 17412  df-prf 17413
This theorem is referenced by: (None)
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