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Theorem natpropd 16683
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 16607 . . 3 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109adantr 480 . . 3 ((𝜑𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
11 nfv 1883 . . . 4 𝑟(𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
12 nfcsb1v 3582 . . . . 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 6241 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) → (1st𝑓) ∈ V)
15 nfv 1883 . . . . . 6 𝑠((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓))
16 nfcsb1v 3582 . . . . . . 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 6241 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) → (1st𝑔) ∈ V)
19 eqid 2651 . . . . . . . . . . 11 (Base‘𝐶) = (Base‘𝐶)
20 eqid 2651 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
21 eqid 2651 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
223ad4antr 769 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
23 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
24 simplr 807 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟 = (1st𝑓))
25 relfunc 16569 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
26 simpllr 815 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)))
2726simpld 474 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑓 ∈ (𝐴 Func 𝐶))
28 1st2ndbr 7261 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
2925, 27, 28sylancr 696 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
3024, 29eqbrtrd 4707 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟(𝐴 Func 𝐶)(2nd𝑓))
3123, 19, 30funcf1 16573 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑟:(Base‘𝐴)⟶(Base‘𝐶))
3231ffvelrnda 6399 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑟𝑥) ∈ (Base‘𝐶))
33 simpr 476 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠 = (1st𝑔))
3426simprd 478 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑔 ∈ (𝐴 Func 𝐶))
35 1st2ndbr 7261 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3625, 34, 35sylancr 696 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
3733, 36eqbrtrd 4707 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠(𝐴 Func 𝐶)(2nd𝑔))
3823, 19, 37funcf1 16573 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → 𝑠:(Base‘𝐴)⟶(Base‘𝐶))
3938ffvelrnda 6399 . . . . . . . . . . 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 16404 . . . . . . . . . 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 7965 . . . . . . . . 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 16403 . . . . . . . . . . 11 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
4342ad3antrrr 766 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶))) ∧ 𝑟 = (1st𝑓)) ∧ 𝑠 = (1st𝑔)) → (Base‘𝐴) = (Base‘𝐵))
4443ixpeq1d 7962 . . . . . . . . 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 2685 . . . . . . . 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 6229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑟𝑥) = (𝑟𝑧))
47 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑠𝑥) = (𝑠𝑧))
4846, 47oveq12d 6708 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)))
4948cbvixpv 7968 . . . . . . . . . 10 X𝑥 ∈ (Base‘𝐴)((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)) = X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧))
5049eleq2i 2722 . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (Hom ‘𝐴) = (Hom ‘𝐴)
54 eqid 2651 . . . . . . . . . . . . 13 (Hom ‘𝐵) = (Hom ‘𝐵)
551ad6antr 777 . . . . . . . . . . . . 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 807 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . 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 16404 . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . 14 (comp‘𝐶) = (comp‘𝐶)
60 eqid 2651 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
613ad7antr 781 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . . 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 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))
6838ad2antrr 762 . . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . . 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 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))
7130ad3antrrr 766 . . . . . . . . . . . . . . . 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 16575 . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑟𝑧) = (𝑟𝑦))
76 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑠𝑧) = (𝑠𝑦))
7775, 76oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7877fvixp 7955 . . . . . . . . . . . . . . . 16 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
7974, 78sylan 487 . . . . . . . . . . . . . . 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 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)) → (𝑎𝑦) ∈ ((𝑟𝑦)(Hom ‘𝐶)(𝑠𝑦)))
8119, 20, 59, 60, 61, 62, 64, 67, 70, 73, 80comfeqval 16415 . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑟𝑧) = (𝑟𝑥))
85 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑠𝑧) = (𝑠𝑥))
8684, 85oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) = ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8786fvixp 7955 . . . . . . . . . . . . . . . 16 ((𝑎X𝑧 ∈ (Base‘𝐴)((𝑟𝑧)(Hom ‘𝐶)(𝑠𝑧)) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ ((𝑟𝑥)(Hom ‘𝐶)(𝑠𝑥)))
8887adantll 750 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . 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 16575 . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 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 16415 . . . . . . . . . . . . 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 2666 . . . . . . . . . . . 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 3184 . . . . . . . . . . 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 3184 . . . . . . . . . 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 3184 . . . . . . . . 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 491 . . . . . . . 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 3227 . . . . . . 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 3575 . . . . . . . 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 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))})
10299, 101eqtrd 2685 . . . . . 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 3587 . . . . 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 3575 . . . . . 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 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))})
106103, 105eqtrd 2685 . . . 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 3587 . . 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 6758 . 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 2651 . . 3 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
110109, 23, 53, 20, 59natfval 16653 . 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 2651 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
112 eqid 2651 . . 3 (Base‘𝐵) = (Base‘𝐵)
113111, 112, 54, 21, 60natfval 16653 . 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 2710 1 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  {crab 2945  Vcvv 3231  csb 3566  cop 4216   class class class wbr 4685  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Homf chomf 16374  compfccomf 16375   Func cfunc 16561   Nat cnat 16648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-cat 16376  df-cid 16377  df-homf 16378  df-comf 16379  df-func 16565  df-nat 16650
This theorem is referenced by:  fucpropd  16684
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