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Theorem natpropd 16837
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 16761 . . 3 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109adantr 472 . . 3 ((𝜑𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11 nfv 1992 . . . 4 𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12 nfcsb1v 3690 . . . . 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 6364 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) ∈ V)
15 nfv 1992 . . . . . 6 𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓))
16 nfcsb1v 3690 . . . . . . 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 6364 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) ∈ V)
19 eqid 2760 . . . . . . . . . . 11 (Base‘𝐶) = (Base‘𝐶)
20 eqid 2760 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
21 eqid 2760 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
223ad4antr 771 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
23 eqid 2760 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
24 simplr 809 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟 = (1st𝑓))
25 relfunc 16723 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
26 simpllr 817 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
2726simpld 477 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑓 ∈ (𝐴 Func 𝐶))
28 1st2ndbr 7384 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
2925, 27, 28sylancr 698 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
3024, 29eqbrtrd 4826 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
3123, 19, 30funcf1 16727 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
3231ffvelrnda 6522 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟𝑥) ∈ (Base‘𝐶))
33 simpr 479 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠 = (1st𝑔))
3426simprd 482 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑔 ∈ (𝐴 Func 𝐶))
35 1st2ndbr 7384 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3625, 34, 35sylancr 698 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3733, 36eqbrtrd 4826 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
3823, 19, 37funcf1 16727 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
3938ffvelrnda 6522 . . . . . . . . . . 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 16558 . . . . . . . . . 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 8089 . . . . . . . . 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 16557 . . . . . . . . . . 11 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
4342ad3antrrr 768 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (Base‘𝐴) = (Base‘𝐵))
4443ixpeq1d 8086 . . . . . . . . 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 2794 . . . . . . . 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 6352 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑟𝑥) = (𝑟𝑧))
47 fveq2 6352 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑠𝑥) = (𝑠𝑧))
4846, 47oveq12d 6831 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
4948cbvixpv 8092 . . . . . . . . . 10 X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))
5049eleq2i 2831 . . . . . . . . 9 (𝑎X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) ↔ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
5143adantr 472 . . . . . . . . . 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 472 . . . . . . . . . . 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 2760 . . . . . . . . . . . . 13 (Hom ‘𝐴) = (Hom ‘𝐴)
54 eqid 2760 . . . . . . . . . . . . 13 (Hom ‘𝐵) = (Hom ‘𝐵)
551ad6antr 779 . . . . . . . . . . . . 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 809 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 16558 . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . 14 (comp‘𝐶) = (comp‘𝐶)
60 eqid 2760 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
613ad7antr 783 . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . 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))
6332adantlr 753 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟𝑥) ∈ (Base‘𝐶))
6463ad2antrr 764 . . . . . . . . . . . . . 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))
6531ad2antrr 764 . . . . . . . . . . . . . . . 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))
6665ffvelrnda 6522 . . . . . . . . . . . . . . 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))
6766adantr 472 . . . . . . . . . . . . . 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))
6838ad2antrr 764 . . . . . . . . . . . . . . . 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))
6968ffvelrnda 6522 . . . . . . . . . . . . . . 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))
7069adantr 472 . . . . . . . . . . . . . 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))
7130ad3antrrr 768 . . . . . . . . . . . . . . . 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))
7223, 53, 20, 71, 56, 57funcf2 16729 . . . . . . . . . . . . . . 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)))
7372ffvelrnda 6522 . . . . . . . . . . . . . 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)))
74 simplr 809 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
75 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑟𝑧) = (𝑟𝑦))
76 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑠𝑧) = (𝑠𝑦))
7775, 76oveq12d 6831 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7877fvixp 8079 . . . . . . . . . . . . . . . 16 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7974, 78sylan 489 . . . . . . . . . . . . . . 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)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
8079adantr 472 . . . . . . . . . . . . . 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)))
8119, 20, 59, 60, 61, 62, 64, 67, 70, 73, 80comfeqval 16569 . . . . . . . . . . . . 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)‘)))
8239adantlr 753 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑠𝑥) ∈ (Base‘𝐶))
8382ad2antrr 764 . . . . . . . . . . . . . 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))
84 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑟𝑧) = (𝑟𝑥))
85 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑠𝑧) = (𝑠𝑥))
8684, 85oveq12d 6831 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8786fvixp 8079 . . . . . . . . . . . . . . . 16 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8887adantll 752 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8988ad2antrr 764 . . . . . . . . . . . . . 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)))
9037ad3antrrr 768 . . . . . . . . . . . . . . . 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))
9123, 53, 20, 90, 56, 57funcf2 16729 . . . . . . . . . . . . . . 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)))
9291ffvelrnda 6522 . . . . . . . . . . . . . 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)))
9319, 20, 59, 60, 61, 62, 64, 83, 70, 89, 92comfeqval 16569 . . . . . . . . . . . . 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)))
9481, 93eqeq12d 2775 . . . . . . . . . . . 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))))
9558, 94raleqbidva 3293 . . . . . . . . . . 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))))
9652, 95raleqbidva 3293 . . . . . . . . . 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))))
9751, 96raleqbidva 3293 . . . . . . . . 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))))
9850, 97sylan2b 493 . . . . . . . 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))))
9945, 98rabeqbidva 3336 . . . . . . 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))})
100 csbeq1a 3683 . . . . . . . 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))})
101100adantl 473 . . . . . . 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))})
10299, 101eqtrd 2794 . . . . . 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))})
10315, 17, 18, 102csbiedf 3695 . . . . 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))})
104 csbeq1a 3683 . . . . . 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))})
105104adantl 473 . . . . 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))})
106103, 105eqtrd 2794 . . . 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))})
10711, 13, 14, 106csbiedf 3695 . . 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))})
1089, 10, 107mpt2eq123dva 6881 . 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))}))
109 eqid 2760 . . 3 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
110109, 23, 53, 20, 59natfval 16807 . 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))})
111 eqid 2760 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
112 eqid 2760 . . 3 (Base‘𝐵) = (Base‘𝐵)
113111, 112, 54, 21, 60natfval 16807 . 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))})
114108, 110, 1133eqtr4g 2819 1 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wnfc 2889  wral 3050  {crab 3054  Vcvv 3340  csb 3674  cop 4327   class class class wbr 4804  Rel wrel 5271  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  Xcixp 8074  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  Homf chomf 16528  compfccomf 16529   Func cfunc 16715   Nat cnat 16802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-ixp 8075  df-cat 16530  df-cid 16531  df-homf 16532  df-comf 16533  df-func 16719  df-nat 16804
This theorem is referenced by:  fucpropd  16838
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