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Theorem natpropd 17977
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
fucpropd.1 (𝜑 → (Homf𝐴) = (Homf𝐵))
fucpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
fucpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
fucpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
fucpropd.a (𝜑𝐴 ∈ Cat)
fucpropd.b (𝜑𝐵 ∈ Cat)
fucpropd.c (𝜑𝐶 ∈ Cat)
fucpropd.d (𝜑𝐷 ∈ Cat)
Assertion
Ref Expression
natpropd (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))

Proof of Theorem natpropd
Dummy variables 𝑎 𝑓 𝑔 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucpropd.1 . . . 4 (𝜑 → (Homf𝐴) = (Homf𝐵))
2 fucpropd.2 . . . 4 (𝜑 → (compf𝐴) = (compf𝐵))
3 fucpropd.3 . . . 4 (𝜑 → (Homf𝐶) = (Homf𝐷))
4 fucpropd.4 . . . 4 (𝜑 → (compf𝐶) = (compf𝐷))
5 fucpropd.a . . . 4 (𝜑𝐴 ∈ Cat)
6 fucpropd.b . . . 4 (𝜑𝐵 ∈ Cat)
7 fucpropd.c . . . 4 (𝜑𝐶 ∈ Cat)
8 fucpropd.d . . . 4 (𝜑𝐷 ∈ Cat)
91, 2, 3, 4, 5, 6, 7, 8funcpropd 17898 . . 3 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109adantr 479 . . 3 ((𝜑𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11 nfv 1909 . . . 4 𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12 nfcsb1v 3919 . . . . 5 𝑟(1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))}
1312a1i 11 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → 𝑟(1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
14 fvexd 6917 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) ∈ V)
15 nfv 1909 . . . . . 6 𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓))
16 nfcsb1v 3919 . . . . . . 7 𝑠(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))}
1716a1i 11 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → 𝑠(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
18 fvexd 6917 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) ∈ V)
19 eqid 2728 . . . . . . . . . . 11 (Base‘𝐶) = (Base‘𝐶)
20 eqid 2728 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
21 eqid 2728 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
223ad4antr 730 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
23 eqid 2728 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
24 simplr 767 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟 = (1st𝑓))
25 relfunc 17857 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
26 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
2726simpld 493 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑓 ∈ (𝐴 Func 𝐶))
28 1st2ndbr 8054 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
2925, 27, 28sylancr 585 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
3024, 29eqbrtrd 5174 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
3123, 19, 30funcf1 17861 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
3231ffvelcdmda 7099 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟𝑥) ∈ (Base‘𝐶))
33 simpr 483 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠 = (1st𝑔))
3426simprd 494 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑔 ∈ (𝐴 Func 𝐶))
35 1st2ndbr 8054 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3625, 34, 35sylancr 585 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3733, 36eqbrtrd 5174 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
3823, 19, 37funcf1 17861 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
3938ffvelcdmda 7099 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑠𝑥) ∈ (Base‘𝐶))
4019, 20, 21, 22, 32, 39homfeqval 17686 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)))
4140ixpeq2dva 8939 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)))
421homfeqbas 17685 . . . . . . . . . . 11 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
4342ad3antrrr 728 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (Base‘𝐴) = (Base‘𝐵))
4443ixpeq1d 8936 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) = X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)))
4541, 44eqtrd 2768 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)))
46 fveq2 6902 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑟𝑥) = (𝑟𝑧))
47 fveq2 6902 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑠𝑥) = (𝑠𝑧))
4846, 47oveq12d 7444 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
4948cbvixpv 8942 . . . . . . . . . 10 X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))
5049eleq2i 2821 . . . . . . . . 9 (𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ↔ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
5143adantr 479 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) → (Base‘𝐴) = (Base‘𝐵))
5251adantr 479 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
53 eqid 2728 . . . . . . . . . . . . 13 (Hom ‘𝐴) = (Hom ‘𝐴)
54 eqid 2728 . . . . . . . . . . . . 13 (Hom ‘𝐵) = (Hom ‘𝐵)
551ad6antr 734 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf𝐴) = (Homf𝐵))
56 simplr 767 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
57 simpr 483 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
5823, 53, 54, 55, 56, 57homfeqval 17686 . . . . . . . . . . . 12 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
59 eqid 2728 . . . . . . . . . . . . . 14 (comp‘𝐶) = (comp‘𝐶)
60 eqid 2728 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
613ad7antr 736 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (Homf𝐶) = (Homf𝐷))
624ad7antr 736 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (compf𝐶) = (compf𝐷))
6332ad5ant13 755 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑟𝑥) ∈ (Base‘𝐶))
6431ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
6564ffvelcdmda 7099 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑟𝑦) ∈ (Base‘𝐶))
6665adantr 479 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑟𝑦) ∈ (Base‘𝐶))
6738ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
6867ffvelcdmda 7099 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑠𝑦) ∈ (Base‘𝐶))
6968adantr 479 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑠𝑦) ∈ (Base‘𝐶))
7030ad3antrrr 728 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
7123, 53, 20, 70, 56, 57funcf2 17863 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(2nd𝑓)𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝑟𝑥)(Hom ‘𝐶)(𝑟𝑦)))
7271ffvelcdmda 7099 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑥(2nd𝑓)𝑦)‘) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑟𝑦)))
73 fveq2 6902 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑟𝑧) = (𝑟𝑦))
74 fveq2 6902 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑠𝑧) = (𝑠𝑦))
7573, 74oveq12d 7444 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7675fvixp 8929 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7776ad5ant24 759 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7819, 20, 59, 60, 61, 62, 63, 66, 69, 72, 77comfeqval 17697 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)))
7939ad5ant13 755 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑠𝑥) ∈ (Base‘𝐶))
80 fveq2 6902 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑟𝑧) = (𝑟𝑥))
81 fveq2 6902 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑠𝑧) = (𝑠𝑥))
8280, 81oveq12d 7444 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8382fvixp 8929 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8483ad5ant23 758 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8537ad3antrrr 728 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
8623, 53, 20, 85, 56, 57funcf2 17863 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑥(2nd𝑔)𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝑠𝑥)(Hom ‘𝐶)(𝑠𝑦)))
8786ffvelcdmda 7099 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → ((𝑥(2nd𝑔)𝑦)‘) ∈ ((𝑠𝑥)(Hom ‘𝐶)(𝑠𝑦)))
8819, 20, 59, 60, 61, 62, 63, 79, 69, 84, 87comfeqval 17697 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥)))
8978, 88eqeq12d 2744 . . . . . . . . . . . 12 ((((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) ∧ ∈ (𝑥(Hom ‘𝐴)𝑦)) → (((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) ↔ ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))))
9058, 89raleqbidva 3325 . . . . . . . . . . 11 (((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) ↔ ∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))))
9152, 90raleqbidva 3325 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))))
9251, 91raleqbidva 3325 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))))
9350, 92sylan2b 592 . . . . . . . 8 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥))) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))))
9445, 93rabeqbidva 3447 . . . . . . 7 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → {𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))} = {𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
95 csbeq1a 3908 . . . . . . . 8 (𝑠 = (1st𝑔) → {𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} = (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
9695adantl 480 . . . . . . 7 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → {𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} = (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
9794, 96eqtrd 2768 . . . . . 6 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → {𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))} = (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
9815, 17, 18, 97csbiedf 3925 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))} = (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
99 csbeq1a 3908 . . . . . 6 (𝑟 = (1st𝑓) → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} = (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
10099adantl 480 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} = (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
10198, 100eqtrd 2768 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))} = (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
10211, 13, 14, 101csbiedf 3925 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))} = (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
1039, 10, 102mpoeq123dva 7501 . 2 (𝜑 → (𝑓 ∈ (𝐴 Func 𝐶), 𝑔 ∈ (𝐴 Func 𝐶) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))}) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑔 ∈ (𝐵 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))}))
104 eqid 2728 . . 3 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
105104, 23, 53, 20, 59natfval 17945 . 2 (𝐴 Nat 𝐶) = (𝑓 ∈ (𝐴 Func 𝐶), 𝑔 ∈ (𝐴 Func 𝐶) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)∀ ∈ (𝑥(Hom ‘𝐴)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐶)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐶)(𝑠𝑦))(𝑎𝑥))})
106 eqid 2728 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
107 eqid 2728 . . 3 (Base‘𝐵) = (Base‘𝐵)
108106, 107, 54, 21, 60natfval 17945 . 2 (𝐵 Nat 𝐷) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑔 ∈ (𝐵 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐵)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)∀ ∈ (𝑥(Hom ‘𝐵)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
109103, 105, 1083eqtr4g 2793 1 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wnfc 2879  wral 3058  {crab 3430  Vcvv 3473  csb 3894  cop 4638   class class class wbr 5152  Rel wrel 5687  wf 6549  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7999  2nd c2nd 8000  Xcixp 8924  Basecbs 17189  Hom chom 17253  compcco 17254  Catccat 17653  Homf chomf 17655  compfccomf 17656   Func cfunc 17849   Nat cnat 17940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-map 8855  df-ixp 8925  df-cat 17657  df-cid 17658  df-homf 17659  df-comf 17660  df-func 17853  df-nat 17942
This theorem is referenced by:  fucpropd  17978
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