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Theorem fuccocl 17063
 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 2795 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2795 . . . 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 17061 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
9 eqid 2795 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
10 eqid 2795 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
112natrcl 17049 . . . . . . . . . . 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 495 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
14 funcrcl 16962 . . . . . . . . 9 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1513, 14syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1615simprd 496 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1716adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18 relfunc 16961 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
19 1st2ndbr 7597 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 13, 19sylancr 587 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
213, 9, 20funcf1 16965 . . . . . . 7 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelrnda 6716 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
232natrcl 17049 . . . . . . . . . . 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 495 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
26 1st2ndbr 7597 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2718, 25, 26sylancr 587 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
283, 9, 27funcf1 16965 . . . . . . 7 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2928ffvelrnda 6716 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3024simprd 496 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
31 1st2ndbr 7597 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3218, 30, 31sylancr 587 . . . . . . . 8 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
333, 9, 32funcf1 16965 . . . . . . 7 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3433ffvelrnda 6716 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
352, 6nat1st2nd 17050 . . . . . . . 8 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3635adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
37 simpr 485 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
382, 36, 3, 10, 37natcl 17052 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
392, 7nat1st2nd 17050 . . . . . . . 8 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
4039adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
412, 40, 3, 10, 37natcl 17052 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
429, 10, 4, 17, 22, 29, 34, 38, 41catcocl 16785 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4342ralrimiva 3149 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
44 fvex 6551 . . . . 5 (Base‘𝐶) ∈ V
45 mptelixpg 8347 . . . . 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 235 . . 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 2883 . 2 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4916adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
5021adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
51 simpr1 1187 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
5250, 51ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
53 simpr2 1188 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
5450, 53ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
5528adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
5655, 53ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
57 eqid 2795 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5820adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
593, 57, 10, 58, 51, 53funcf2 16967 . . . . . . 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 1189 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
6159, 60ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
6235adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
632, 62, 3, 10, 53natcl 17052 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
6433adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
6564, 53ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐻)‘𝑦) ∈ (Base‘𝐷))
6639adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
672, 66, 3, 10, 53natcl 17052 . . . . . 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 16786 . . . . 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 17054 . . . . . . 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 7032 . . . . . 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 6717 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
722, 62, 3, 10, 51natcl 17052 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
7327adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
743, 57, 10, 73, 51, 53funcf2 16967 . . . . . . . 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 6717 . . . . . . 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 16786 . . . . . 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 17054 . . . . . . 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 7031 . . . . . 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 2837 . . . . 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 6717 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
812, 66, 3, 10, 51natcl 17052 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
8232adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
833, 57, 10, 82, 51, 53funcf2 16967 . . . . . . 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 6717 . . . . . 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 16786 . . . . 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 2835 . . . 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 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑅 ∈ (𝐹𝑁𝐺))
887adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑆 ∈ (𝐺𝑁𝐻))
891, 2, 3, 4, 5, 87, 88, 53fuccoval 17062 . . . . 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 7031 . . . 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 17062 . . . . 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 7032 . . . 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 2841 . . 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 3159 . 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 17047 . 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 709 1 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  Vcvv 3437  ⟨cop 4478   class class class wbr 4962   ↦ cmpt 5041  Rel wrel 5448  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  Xcixp 8310  Basecbs 16312  Hom chom 16405  compcco 16406  Catccat 16764   Func cfunc 16953   Nat cnat 17040   FuncCat cfuc 17041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-hom 16418  df-cco 16419  df-cat 16768  df-func 16957  df-nat 17042  df-fuc 17043 This theorem is referenced by:  fucass  17067  fuccatid  17068  evlfcllem  17300  yonedalem3b  17358
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