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Theorem isnat 16828
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 16827 . . . . 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 6365 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) ∈ V)
9 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → 𝑓 = ⟨𝐹, 𝐺⟩)
109fveq2d 6357 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = (1st ‘⟨𝐹, 𝐺⟩))
11 relfunc 16743 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
12 isnat.f . . . . . . . . 9 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
13 brrelex12 5312 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1411, 12, 13sylancr 698 . . . . . . . 8 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
15 op1stg 7346 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1614, 15syl 17 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1716adantr 472 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1810, 17eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = 𝐹)
19 fvexd 6365 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) ∈ V)
20 simplrr 820 . . . . . . . 8 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → 𝑔 = ⟨𝐾, 𝐿⟩)
2120fveq2d 6357 . . . . . . 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 5312 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐾(𝐶 Func 𝐷)𝐿) → (𝐾 ∈ V ∧ 𝐿 ∈ V))
2411, 22, 23sylancr 698 . . . . . . . . 9 (𝜑 → (𝐾 ∈ V ∧ 𝐿 ∈ V))
25 op1stg 7346 . . . . . . . . 9 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2624, 25syl 17 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2726ad2antrr 764 . . . . . . 7 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2821, 27eqtrd 2794 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) = 𝐾)
29 simplr 809 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑟 = 𝐹)
3029fveq1d 6355 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑥) = (𝐹𝑥))
31 simpr 479 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
3231fveq1d 6355 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑥) = (𝐾𝑥))
3330, 32oveq12d 6832 . . . . . . . 8 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑟𝑥)𝐽(𝑠𝑥)) = ((𝐹𝑥)𝐽(𝐾𝑥)))
3433ixpeq2dv 8092 . . . . . . 7 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) = X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
3529fveq1d 6355 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑦) = (𝐹𝑦))
3630, 35opeq12d 4561 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑟𝑦)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3731fveq1d 6355 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑦) = (𝐾𝑦))
3836, 37oveq12d 6832 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦)))
39 eqidd 2761 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑦) = (𝑎𝑦))
409ad2antrr 764 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑓 = ⟨𝐹, 𝐺⟩)
4140fveq2d 6357 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = (2nd ‘⟨𝐹, 𝐺⟩))
42 op2ndg 7347 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4314, 42syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4443ad3antrrr 768 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4541, 44eqtrd 2794 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = 𝐺)
4645oveqd 6831 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑓)𝑦) = (𝑥𝐺𝑦))
4746fveq1d 6355 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥𝐺𝑦)‘))
4838, 39, 47oveq123d 6835 . . . . . . . . . 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 4561 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑠𝑥)⟩ = ⟨(𝐹𝑥), (𝐾𝑥)⟩)
5049, 37oveq12d 6832 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦)))
5120adantr 472 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑔 = ⟨𝐾, 𝐿⟩)
5251fveq2d 6357 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = (2nd ‘⟨𝐾, 𝐿⟩))
53 op2ndg 7347 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5424, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5554ad3antrrr 768 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5652, 55eqtrd 2794 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = 𝐿)
5756oveqd 6831 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑔)𝑦) = (𝑥𝐿𝑦))
5857fveq1d 6355 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥𝐿𝑦)‘))
59 eqidd 2761 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑥) = (𝑎𝑥))
6050, 58, 59oveq123d 6835 . . . . . . . . . 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 2775 . . . . . . . . 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 3124 . . . . . . . 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 3127 . . . . . . 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 3335 . . . . . 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 3702 . . . . 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 3702 . . . 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 4805 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
6812, 67sylib 208 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
69 df-br 4805 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
7022, 69sylib 208 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
71 ovex 6842 . . . . . . . 8 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7271rgenw 3062 . . . . . . 7 𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
73 ixpexg 8100 . . . . . . 7 (∀𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V → X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V)
7472, 73ax-mp 5 . . . . . 6 X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7574rabex 4964 . . . . 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, 76ovmpt2d 6954 . . 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 6352 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑦) = (𝐴𝑦))
8079oveq1d 6829 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)))
81 fveq1 6352 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑥) = (𝐴𝑥))
8281oveq2d 6830 . . . . . 6 (𝑎 = 𝐴 → (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))
8380, 82eqeq12d 2775 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) ↔ ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
8483ralbidv 3124 . . . 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 3127 . . 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 3504 . 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, 86syl6bb 276 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  csb 3674  cop 4327   class class class wbr 4804  Rel wrel 5271  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332  2nd c2nd 7333  Xcixp 8076  Basecbs 16079  Hom chom 16174  compcco 16175   Func cfunc 16735   Nat cnat 16822
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 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  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-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-ixp 8077  df-func 16739  df-nat 16824
This theorem is referenced by:  isnat2  16829  natixp  16833  nati  16836
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