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Theorem 1st2ndprf 17308
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 2772 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 1st2ndprf.t . . . . . . 7 𝑇 = (𝐷 ×c 𝐸)
3 eqid 2772 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
4 eqid 2772 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
52, 3, 4xpcbas 17280 . . . . . 6 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
6 relfunc 16984 . . . . . . 7 Rel (𝐶 Func 𝑇)
7 1st2ndprf.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝑇))
8 1st2ndbr 7547 . . . . . . 7 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
96, 7, 8sylancr 578 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
101, 5, 9funcf1 16988 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
1110feqmptd 6556 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
1210ffvelrnda 6670 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
13 1st2nd2 7534 . . . . . . 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 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝑇))
16 1st2ndprf.d . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
17 1st2ndprf.e . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
18 eqid 2772 . . . . . . . . . . 11 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
192, 16, 17, 181stfcl 17299 . . . . . . . . . 10 (𝜑 → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
2019adantr 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
21 simpr 477 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
221, 15, 20, 21cofu1 17006 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)))
23 eqid 2772 . . . . . . . . 9 (Hom ‘𝑇) = (Hom ‘𝑇)
2416adantr 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2517adantr 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
262, 5, 23, 24, 25, 18, 121stf1 17294 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘((1st𝐹)‘𝑥)) = (1st ‘((1st𝐹)‘𝑥)))
2722, 26eqtrd 2808 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥) = (1st ‘((1st𝐹)‘𝑥)))
28 eqid 2772 . . . . . . . . . . 11 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
292, 16, 17, 282ndfcl 17300 . . . . . . . . . 10 (𝜑 → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
3029adantr 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐷 2ndF 𝐸) ∈ (𝑇 Func 𝐸))
311, 15, 30, 21cofu1 17006 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)))
322, 5, 23, 24, 25, 28, 122ndf1 17297 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘((1st𝐹)‘𝑥)) = (2nd ‘((1st𝐹)‘𝑥)))
3331, 32eqtrd 2808 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥) = (2nd ‘((1st𝐹)‘𝑥)))
3427, 33opeq12d 4679 . . . . . 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 2811 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) = ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩)
3635mpteq2dva 5016 . . . 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 2808 . . 3 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘((𝐷 1stF 𝐸) ∘func 𝐹))‘𝑥), ((1st ‘((𝐷 2ndF 𝐸) ∘func 𝐹))‘𝑥)⟩))
381, 9funcfn2 16991 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
39 fnov 7092 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4038, 39sylib 210 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
41 eqid 2772 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
429adantr 473 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝑇)(2nd𝐹))
43 simprl 758 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
44 simprr 760 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
451, 41, 23, 42, 43, 44funcf2 16990 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦)))
4645feqmptd 6556 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
472, 23relxpchom 17283 . . . . . . . . . 10 Rel (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))
4845ffvelrnda 6670 . . . . . . . . . 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 7544 . . . . . . . . . 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 578 . . . . . . . . 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 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝑇))
5219ad2antrr 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐷 1stF 𝐸) ∈ (𝑇 Func 𝐷))
5343adantr 473 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
5444adantr 473 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
55 simpr 477 . . . . . . . . . . . 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 17008 . . . . . . . . . . 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 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
5817adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐸 ∈ Cat)
5912adantrr 704 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6010ffvelrnda 6670 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrl 703 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
622, 5, 23, 57, 58, 18, 59, 611stf2 17295 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝐹)‘𝑦)) = (1st ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
6362adantr 473 . . . . . . . . . . . 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 6495 . . . . . . . . . . 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 6513 . . . . . . . . . . 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 2812 . . . . . . . . . 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 713 . . . . . . . . . . . 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 17008 . . . . . . . . . . 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 17298 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝐹)‘𝑦)) = (2nd ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑇)((1st𝐹)‘𝑦))))
7069adantr 473 . . . . . . . . . . . 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 6495 . . . . . . . . . . 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 6513 . . . . . . . . . . 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 2812 . . . . . . . . . 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 4679 . . . . . . . . 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 2811 . . . . . . . 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 5016 . . . . . . 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 2808 . . . . . 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 1095 . . . . 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 7043 . . . 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 2808 . . 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 4679 . 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 7544 . . 3 ((Rel (𝐶 Func 𝑇) ∧ 𝐹 ∈ (𝐶 Func 𝑇)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
836, 7, 82sylancr 578 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
84 eqid 2772 . . 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 17010 . . 3 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐷))
867, 29cofucl 17010 . . 3 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝐹) ∈ (𝐶 Func 𝐸))
8784, 1, 41, 85, 86prfval 17301 . 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 2818 1 (𝜑𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  cop 4441   class class class wbr 4923  cmpt 5002   × cxp 5399  cres 5403  Rel wrel 5406   Fn wfn 6177  cfv 6182  (class class class)co 6970  cmpo 6972  1st c1st 7493  2nd c2nd 7494  Basecbs 16333  Hom chom 16426  Catccat 16787   Func cfunc 16976  func ccofu 16978   ×c cxpc 17270   1stF c1stf 17271   2ndF c2ndf 17272   ⟨,⟩F cprf 17273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-ixp 8254  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-3 11498  df-4 11499  df-5 11500  df-6 11501  df-7 11502  df-8 11503  df-9 11504  df-n0 11702  df-z 11788  df-dec 11906  df-uz 12053  df-fz 12703  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-hom 16439  df-cco 16440  df-cat 16791  df-cid 16792  df-func 16980  df-cofu 16982  df-xpc 17274  df-1stf 17275  df-2ndf 17276  df-prf 17277
This theorem is referenced by: (None)
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