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Theorem natpropd 18024
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 17947 . . 3 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109adantr 480 . . 3 ((𝜑𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11 nfv 1914 . . . 4 𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12 nfcsb1v 3923 . . . . 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 6921 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) ∈ V)
15 nfv 1914 . . . . . 6 𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓))
16 nfcsb1v 3923 . . . . . . 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 6921 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) ∈ V)
19 eqid 2737 . . . . . . . . . . 11 (Base‘𝐶) = (Base‘𝐶)
20 eqid 2737 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
21 eqid 2737 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
223ad4antr 732 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
23 eqid 2737 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
24 simplr 769 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟 = (1st𝑓))
25 relfunc 17907 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
26 simpllr 776 . . . . . . . . . . . . . . . 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 8067 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
2925, 27, 28sylancr 587 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
3024, 29eqbrtrd 5165 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
3123, 19, 30funcf1 17911 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
3231ffvelcdmda 7104 . . . . . . . . . . 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 8067 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3625, 34, 35sylancr 587 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3733, 36eqbrtrd 5165 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
3823, 19, 37funcf1 17911 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
3938ffvelcdmda 7104 . . . . . . . . . . 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 17740 . . . . . . . . . 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 8952 . . . . . . . . 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 17739 . . . . . . . . . . 11 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
4342ad3antrrr 730 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (Base‘𝐴) = (Base‘𝐵))
4443ixpeq1d 8949 . . . . . . . . 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 2777 . . . . . . . 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 6906 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑟𝑥) = (𝑟𝑧))
47 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑠𝑥) = (𝑠𝑧))
4846, 47oveq12d 7449 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
4948cbvixpv 8955 . . . . . . . . . 10 X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))
5049eleq2i 2833 . . . . . . . . 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 2737 . . . . . . . . . . . . 13 (Hom ‘𝐴) = (Hom ‘𝐴)
54 eqid 2737 . . . . . . . . . . . . 13 (Hom ‘𝐵) = (Hom ‘𝐵)
551ad6antr 736 . . . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 17740 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (comp‘𝐶) = (comp‘𝐶)
60 eqid 2737 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
613ad7antr 738 . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . 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 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)) → (𝑟𝑥) ∈ (Base‘𝐶))
6431ad2antrr 726 . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . 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 17913 . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑟𝑧) = (𝑟𝑦))
74 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑠𝑧) = (𝑠𝑦))
7573, 74oveq12d 7449 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7675fvixp 8942 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7776ad5ant24 761 . . . . . . . . . . . . . 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 17751 . . . . . . . . . . . . 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 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)) → (𝑠𝑥) ∈ (Base‘𝐶))
80 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑟𝑧) = (𝑟𝑥))
81 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝑠𝑧) = (𝑠𝑥))
8280, 81oveq12d 7449 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8382fvixp 8942 . . . . . . . . . . . . . . 15 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8483ad5ant23 760 . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . 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 17913 . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . 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 17751 . . . . . . . . . . . . 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 2753 . . . . . . . . . . . 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 3332 . . . . . . . . . . 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 3332 . . . . . . . . . 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 3332 . . . . . . . . 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 594 . . . . . . . 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 3453 . . . . . . 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 3913 . . . . . . . 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 2777 . . . . . 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 3929 . . . . 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 3913 . . . . . 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 2777 . . . 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 3929 . . 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 7507 . 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 2737 . . 3 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
105104, 23, 53, 20, 59natfval 17994 . 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 2737 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
107 eqid 2737 . . 3 (Base‘𝐵) = (Base‘𝐵)
108106, 107, 54, 21, 60natfval 17994 . 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 2802 1 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  {crab 3436  Vcvv 3480  csb 3899  cop 4632   class class class wbr 5143  Rel wrel 5690  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Xcixp 8937  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Homf chomf 17709  compfccomf 17710   Func cfunc 17899   Nat cnat 17989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-func 17903  df-nat 17991
This theorem is referenced by:  fucpropd  18025
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