MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnat Structured version   Visualization version   GIF version

Theorem isnat 18002
Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natfval.1 𝑁 = (𝐶 Nat 𝐷)
natfval.b 𝐵 = (Base‘𝐶)
natfval.h 𝐻 = (Hom ‘𝐶)
natfval.j 𝐽 = (Hom ‘𝐷)
natfval.o · = (comp‘𝐷)
isnat.f (𝜑𝐹(𝐶 Func 𝐷)𝐺)
isnat.g (𝜑𝐾(𝐶 Func 𝐷)𝐿)
Assertion
Ref Expression
isnat (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))))
Distinct variable groups:   𝑥,,𝑦,𝐴   𝑥,𝐵,𝑦   𝐶,,𝑥,𝑦   ,𝐹,𝑥,𝑦   ,𝐺,𝑥,𝑦   ,𝐻   𝜑,,𝑥,𝑦   ,𝐾,𝑥,𝑦   ,𝐿,𝑥,𝑦   𝐷,,𝑥,𝑦
Allowed substitution hints:   𝐵()   · (𝑥,𝑦,)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,)   𝑁(𝑥,𝑦,)

Proof of Theorem isnat
Dummy variables 𝑎 𝑓 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natfval.1 . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
2 natfval.b . . . . . 6 𝐵 = (Base‘𝐶)
3 natfval.h . . . . . 6 𝐻 = (Hom ‘𝐶)
4 natfval.j . . . . . 6 𝐽 = (Hom ‘𝐷)
5 natfval.o . . . . . 6 · = (comp‘𝐷)
61, 2, 3, 4, 5natfval 18001 . . . . 5 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
76a1i 11 . . . 4 (𝜑𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
8 fvexd 6922 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) ∈ V)
9 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → 𝑓 = ⟨𝐹, 𝐺⟩)
109fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = (1st ‘⟨𝐹, 𝐺⟩))
11 relfunc 17913 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
12 isnat.f . . . . . . . . 9 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
13 brrelex12 5741 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1411, 12, 13sylancr 587 . . . . . . . 8 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
15 op1stg 8025 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1614, 15syl 17 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1716adantr 480 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1810, 17eqtrd 2775 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = 𝐹)
19 fvexd 6922 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) ∈ V)
20 simplrr 778 . . . . . . . 8 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → 𝑔 = ⟨𝐾, 𝐿⟩)
2120fveq2d 6911 . . . . . . 7 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) = (1st ‘⟨𝐾, 𝐿⟩))
22 isnat.g . . . . . . . . . 10 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
23 brrelex12 5741 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐾(𝐶 Func 𝐷)𝐿) → (𝐾 ∈ V ∧ 𝐿 ∈ V))
2411, 22, 23sylancr 587 . . . . . . . . 9 (𝜑 → (𝐾 ∈ V ∧ 𝐿 ∈ V))
25 op1stg 8025 . . . . . . . . 9 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2624, 25syl 17 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2726ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2821, 27eqtrd 2775 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) = 𝐾)
29 simplr 769 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑟 = 𝐹)
3029fveq1d 6909 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑥) = (𝐹𝑥))
31 simpr 484 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
3231fveq1d 6909 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑥) = (𝐾𝑥))
3330, 32oveq12d 7449 . . . . . . . 8 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑟𝑥)𝐽(𝑠𝑥)) = ((𝐹𝑥)𝐽(𝐾𝑥)))
3433ixpeq2dv 8952 . . . . . . 7 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) = X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
3529fveq1d 6909 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑦) = (𝐹𝑦))
3630, 35opeq12d 4886 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑟𝑦)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3731fveq1d 6909 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑦) = (𝐾𝑦))
3836, 37oveq12d 7449 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦)))
39 eqidd 2736 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑦) = (𝑎𝑦))
409ad2antrr 726 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑓 = ⟨𝐹, 𝐺⟩)
4140fveq2d 6911 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = (2nd ‘⟨𝐹, 𝐺⟩))
42 op2ndg 8026 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4314, 42syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4443ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4541, 44eqtrd 2775 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = 𝐺)
4645oveqd 7448 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑓)𝑦) = (𝑥𝐺𝑦))
4746fveq1d 6909 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥𝐺𝑦)‘))
4838, 39, 47oveq123d 7452 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = ((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)))
4930, 32opeq12d 4886 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑠𝑥)⟩ = ⟨(𝐹𝑥), (𝐾𝑥)⟩)
5049, 37oveq12d 7449 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦)))
5120adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑔 = ⟨𝐾, 𝐿⟩)
5251fveq2d 6911 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = (2nd ‘⟨𝐾, 𝐿⟩))
53 op2ndg 8026 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5424, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5554ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5652, 55eqtrd 2775 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = 𝐿)
5756oveqd 7448 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑔)𝑦) = (𝑥𝐿𝑦))
5857fveq1d 6909 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥𝐿𝑦)‘))
59 eqidd 2736 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑥) = (𝑎𝑥))
6050, 58, 59oveq123d 7452 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)))
6148, 60eqeq12d 2751 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥)) ↔ ((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))))
6261ralbidv 3176 . . . . . . . 8 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (∀ ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥)) ↔ ∀ ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))))
63622ralbidv 3219 . . . . . . 7 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))))
6434, 63rabeqbidv 3452 . . . . . 6 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → {𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))} = {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))})
6519, 28, 64csbied2 3948 . . . . 5 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))} = {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))})
668, 18, 65csbied2 3948 . . . 4 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))} = {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))})
67 df-br 5149 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
6812, 67sylib 218 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
69 df-br 5149 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
7022, 69sylib 218 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
71 ovex 7464 . . . . . . . 8 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7271rgenw 3063 . . . . . . 7 𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
73 ixpexg 8961 . . . . . . 7 (∀𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V → X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V)
7472, 73ax-mp 5 . . . . . 6 X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7574rabex 5345 . . . . 5 {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))} ∈ V
7675a1i 11 . . . 4 (𝜑 → {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))} ∈ V)
777, 66, 68, 70, 76ovmpod 7585 . . 3 (𝜑 → (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) = {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))})
7877eleq2d 2825 . 2 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ 𝐴 ∈ {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))}))
79 fveq1 6906 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑦) = (𝐴𝑦))
8079oveq1d 7446 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)))
81 fveq1 6906 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑥) = (𝐴𝑥))
8281oveq2d 7447 . . . . . 6 (𝑎 = 𝐴 → (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))
8380, 82eqeq12d 2751 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) ↔ ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
8483ralbidv 3176 . . . 4 (𝑎 = 𝐴 → (∀ ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) ↔ ∀ ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
85842ralbidv 3219 . . 3 (𝑎 = 𝐴 → (∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) ↔ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
8685elrab 3695 . 2 (𝐴 ∈ {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))} ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
8778, 86bitrdi 287 1 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  csb 3908  cop 4637   class class class wbr 5148  Rel wrel 5694  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Xcixp 8936  Basecbs 17245  Hom chom 17309  compcco 17310   Func cfunc 17905   Nat cnat 17996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ixp 8937  df-func 17909  df-nat 17998
This theorem is referenced by:  isnat2  18003  natixp  18007  nati  18010
  Copyright terms: Public domain W3C validator