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Theorem natpropd 17939
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 17860 . . 3 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109adantr 480 . . 3 ((𝜑𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11 nfv 1916 . . . 4 𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12 nfcsb1v 3918 . . . . 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 6906 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) ∈ V)
15 nfv 1916 . . . . . 6 𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓))
16 nfcsb1v 3918 . . . . . . 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 6906 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) ∈ V)
19 eqid 2731 . . . . . . . . . . 11 (Base‘𝐶) = (Base‘𝐶)
20 eqid 2731 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
21 eqid 2731 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
223ad4antr 729 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
23 eqid 2731 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
24 simplr 766 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟 = (1st𝑓))
25 relfunc 17819 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
26 simpllr 773 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
2726simpld 494 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑓 ∈ (𝐴 Func 𝐶))
28 1st2ndbr 8032 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
2925, 27, 28sylancr 586 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
3024, 29eqbrtrd 5170 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
3123, 19, 30funcf1 17823 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
3231ffvelcdmda 7086 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟𝑥) ∈ (Base‘𝐶))
33 simpr 484 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠 = (1st𝑔))
3426simprd 495 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑔 ∈ (𝐴 Func 𝐶))
35 1st2ndbr 8032 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3625, 34, 35sylancr 586 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3733, 36eqbrtrd 5170 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
3823, 19, 37funcf1 17823 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
3938ffvelcdmda 7086 . . . . . . . . . . 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 17648 . . . . . . . . . 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 8912 . . . . . . . . 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 17647 . . . . . . . . . . 11 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
4342ad3antrrr 727 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (Base‘𝐴) = (Base‘𝐵))
4443ixpeq1d 8909 . . . . . . . . 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 2771 . . . . . . . 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 6891 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑟𝑥) = (𝑟𝑧))
47 fveq2 6891 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑠𝑥) = (𝑠𝑧))
4846, 47oveq12d 7430 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
4948cbvixpv 8915 . . . . . . . . . 10 X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))
5049eleq2i 2824 . . . . . . . . 9 (𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ↔ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
5143adantr 480 . . . . . . . . . 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 480 . . . . . . . . . . 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 2731 . . . . . . . . . . . . 13 (Hom ‘𝐴) = (Hom ‘𝐴)
54 eqid 2731 . . . . . . . . . . . . 13 (Hom ‘𝐵) = (Hom ‘𝐵)
551ad6antr 733 . . . . . . . . . . . . 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 766 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 17648 . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . 14 (comp‘𝐶) = (comp‘𝐶)
60 eqid 2731 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
613ad7antr 735 . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 17825 . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑟𝑧) = (𝑟𝑦))
74 fveq2 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑠𝑧) = (𝑠𝑦))
7573, 74oveq12d 7430 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7675fvixp 8902 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7776ad5ant24 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)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7819, 20, 59, 60, 61, 62, 63, 66, 69, 72, 77comfeqval 17659 . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑟𝑧) = (𝑟𝑥))
81 fveq2 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑠𝑧) = (𝑠𝑥))
8280, 81oveq12d 7430 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8382fvixp 8902 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8483ad5ant23 757 . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 17825 . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . 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 17659 . . . . . . . . . . . . 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 2747 . . . . . . . . . . . 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 3326 . . . . . . . . . . 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 3326 . . . . . . . . . 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 3326 . . . . . . . . 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 593 . . . . . . . 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 3907 . . . . . . . 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 481 . . . . . . 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 2771 . . . . . 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 3924 . . . . 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 3907 . . . . . 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 481 . . . . 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 2771 . . . 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 3924 . . 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 7486 . 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 2731 . . 3 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
105104, 23, 53, 20, 59natfval 17907 . 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 2731 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
107 eqid 2731 . . 3 (Base‘𝐵) = (Base‘𝐵)
108106, 107, 54, 21, 60natfval 17907 . 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 2796 1 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wnfc 2882  wral 3060  {crab 3431  Vcvv 3473  csb 3893  cop 4634   class class class wbr 5148  Rel wrel 5681  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  1st c1st 7977  2nd c2nd 7978  Xcixp 8897  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  Homf chomf 17617  compfccomf 17618   Func cfunc 17811   Nat cnat 17902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-ixp 8898  df-cat 17619  df-cid 17620  df-homf 17621  df-comf 17622  df-func 17815  df-nat 17904
This theorem is referenced by:  fucpropd  17940
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