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Theorem fuccocl 16389
Description: The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuccocl.q 𝑄 = (𝐶 FuncCat 𝐷)
fuccocl.n 𝑁 = (𝐶 Nat 𝐷)
fuccocl.x = (comp‘𝑄)
fuccocl.r (𝜑𝑅 ∈ (𝐹𝑁𝐺))
fuccocl.s (𝜑𝑆 ∈ (𝐺𝑁𝐻))
Assertion
Ref Expression
fuccocl (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Proof of Theorem fuccocl
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fuccocl.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
2 fuccocl.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
3 eqid 2605 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2605 . . . 4 (comp‘𝐷) = (comp‘𝐷)
5 fuccocl.x . . . 4 = (comp‘𝑄)
6 fuccocl.r . . . 4 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
7 fuccocl.s . . . 4 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
81, 2, 3, 4, 5, 6, 7fucco 16387 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
9 eqid 2605 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
10 eqid 2605 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
112natrcl 16375 . . . . . . . . . . 11 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
126, 11syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
1312simpld 473 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
14 funcrcl 16288 . . . . . . . . 9 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1513, 14syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1615simprd 477 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1716adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18 relfunc 16287 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
19 1st2ndbr 7081 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 13, 19sylancr 693 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
213, 9, 20funcf1 16291 . . . . . . 7 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelrnda 6248 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
232natrcl 16375 . . . . . . . . . . 11 (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
247, 23syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
2524simpld 473 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
26 1st2ndbr 7081 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2718, 25, 26sylancr 693 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
283, 9, 27funcf1 16291 . . . . . . 7 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2928ffvelrnda 6248 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3024simprd 477 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
31 1st2ndbr 7081 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3218, 30, 31sylancr 693 . . . . . . . 8 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
333, 9, 32funcf1 16291 . . . . . . 7 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3433ffvelrnda 6248 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
352, 6nat1st2nd 16376 . . . . . . . 8 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3635adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
37 simpr 475 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
382, 36, 3, 10, 37natcl 16378 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
392, 7nat1st2nd 16376 . . . . . . . 8 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
4039adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
412, 40, 3, 10, 37natcl 16378 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
429, 10, 4, 17, 22, 29, 34, 38, 41catcocl 16111 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4342ralrimiva 2944 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
44 fvex 6094 . . . . 5 (Base‘𝐶) ∈ V
45 mptelixpg 7804 . . . . 5 ((Base‘𝐶) ∈ V → ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥))))
4644, 45ax-mp 5 . . . 4 ((𝑥 ∈ (Base‘𝐶) ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4743, 46sylibr 222 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
488, 47eqeltrd 2683 . 2 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4916adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
5021adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
51 simpr1 1059 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
5250, 51ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
53 simpr2 1060 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
5450, 53ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
5528adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
5655, 53ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
57 eqid 2605 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5820adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
593, 57, 10, 58, 51, 53funcf2 16293 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
60 simpr3 1061 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
6159, 60ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
6235adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
632, 62, 3, 10, 53natcl 16378 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
6433adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
6564, 53ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐻)‘𝑦) ∈ (Base‘𝐷))
6639adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
672, 66, 3, 10, 53natcl 16378 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐻)‘𝑦)))
689, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67catass 16112 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆𝑦)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = ((𝑆𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑅𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓))))
692, 62, 3, 57, 4, 51, 53, 60nati 16380 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑅𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐺)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑦))(𝑅𝑥)))
7069oveq2d 6539 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑅𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓))) = ((𝑆𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(((𝑥(2nd𝐺)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑦))(𝑅𝑥))))
7155, 51ffvelrnd 6249 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
722, 62, 3, 10, 51natcl 16378 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
7327adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
743, 57, 10, 73, 51, 53funcf2 16293 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
7574, 60ffvelrnd 6249 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
769, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67catass 16112 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆𝑦)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐺)𝑦)‘𝑓))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑥)) = ((𝑆𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(((𝑥(2nd𝐺)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑦))(𝑅𝑥))))
772, 66, 3, 57, 4, 51, 53, 60nati 16380 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆𝑦)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐺)𝑦)‘𝑓)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑆𝑥)))
7877oveq1d 6538 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆𝑦)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐺)𝑦)‘𝑓))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑥)) = ((((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑆𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑥)))
7970, 76, 783eqtr2d 2645 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑅𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓))) = ((((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑆𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑥)))
8064, 51ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
812, 66, 3, 10, 51natcl 16378 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
8232adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
833, 57, 10, 82, 51, 53funcf2 16293 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd𝐻)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑦)))
8483, 60ffvelrnd 6249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd𝐻)𝑦)‘𝑓) ∈ (((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑦)))
859, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84catass 16112 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑆𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑥)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
8668, 79, 853eqtrd 2643 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆𝑦)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
876adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (𝐹𝑁𝐺))
887adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (𝐺𝑁𝐻))
891, 2, 3, 4, 5, 87, 88, 53fuccoval 16388 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑦) = ((𝑆𝑦)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑦)))
9089oveq1d 6538 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑆𝑦)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))(𝑅𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)))
911, 2, 3, 4, 5, 87, 88, 51fuccoval 16388 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)))
9291oveq2d 6539 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
9386, 90, 923eqtr4d 2649 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)))
9493ralrimivvva 2950 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)))
952, 3, 57, 10, 4, 13, 30isnat2 16373 . 2 (𝜑 → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻) ↔ ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐻)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑦))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)))))
9648, 94, 95mpbir2and 958 1 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2891  Vcvv 3168  cop 4126   class class class wbr 4573  cmpt 4633  Rel wrel 5029  wf 5782  cfv 5786  (class class class)co 6523  1st c1st 7030  2nd c2nd 7031  Xcixp 7767  Basecbs 15637  Hom chom 15721  compcco 15722  Catccat 16090   Func cfunc 16279   Nat cnat 16366   FuncCat cfuc 16367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-fz 12149  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-hom 15735  df-cco 15736  df-cat 16094  df-func 16283  df-nat 16368  df-fuc 16369
This theorem is referenced by:  fucass  16393  fuccatid  16394  evlfcllem  16626  yonedalem3b  16684
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