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Theorem isnat 16588
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 16587 . . . . 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 6190 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) ∈ V)
9 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → 𝑓 = ⟨𝐹, 𝐺⟩)
109fveq2d 6182 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = (1st ‘⟨𝐹, 𝐺⟩))
11 relfunc 16503 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
12 isnat.f . . . . . . . . 9 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
13 brrelex12 5145 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1411, 12, 13sylancr 694 . . . . . . . 8 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
15 op1stg 7165 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1614, 15syl 17 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1716adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1810, 17eqtrd 2654 . . . . 5 ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st𝑓) = 𝐹)
19 fvexd 6190 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) ∈ V)
20 simplrr 800 . . . . . . . 8 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → 𝑔 = ⟨𝐾, 𝐿⟩)
2120fveq2d 6182 . . . . . . 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 5145 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐾(𝐶 Func 𝐷)𝐿) → (𝐾 ∈ V ∧ 𝐿 ∈ V))
2411, 22, 23sylancr 694 . . . . . . . . 9 (𝜑 → (𝐾 ∈ V ∧ 𝐿 ∈ V))
25 op1stg 7165 . . . . . . . . 9 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2624, 25syl 17 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2726ad2antrr 761 . . . . . . 7 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
2821, 27eqtrd 2654 . . . . . 6 (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st𝑔) = 𝐾)
29 simplr 791 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑟 = 𝐹)
3029fveq1d 6180 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑥) = (𝐹𝑥))
31 simpr 477 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
3231fveq1d 6180 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑥) = (𝐾𝑥))
3330, 32oveq12d 6653 . . . . . . . 8 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑟𝑥)𝐽(𝑠𝑥)) = ((𝐹𝑥)𝐽(𝐾𝑥)))
3433ixpeq2dv 7909 . . . . . . 7 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) = X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
3529fveq1d 6180 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟𝑦) = (𝐹𝑦))
3630, 35opeq12d 4401 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑟𝑦)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3731fveq1d 6180 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠𝑦) = (𝐾𝑦))
3836, 37oveq12d 6653 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦)))
39 eqidd 2621 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑦) = (𝑎𝑦))
409ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑓 = ⟨𝐹, 𝐺⟩)
4140fveq2d 6182 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = (2nd ‘⟨𝐹, 𝐺⟩))
42 op2ndg 7166 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4314, 42syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4443ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
4541, 44eqtrd 2654 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑓) = 𝐺)
4645oveqd 6652 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑓)𝑦) = (𝑥𝐺𝑦))
4746fveq1d 6180 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥𝐺𝑦)‘))
4838, 39, 47oveq123d 6656 . . . . . . . . . 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 4401 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟𝑥), (𝑠𝑥)⟩ = ⟨(𝐹𝑥), (𝐾𝑥)⟩)
5049, 37oveq12d 6653 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦)) = (⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦)))
5120adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑔 = ⟨𝐾, 𝐿⟩)
5251fveq2d 6182 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = (2nd ‘⟨𝐾, 𝐿⟩))
53 op2ndg 7166 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5424, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5554ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5652, 55eqtrd 2654 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd𝑔) = 𝐿)
5756oveqd 6652 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd𝑔)𝑦) = (𝑥𝐿𝑦))
5857fveq1d 6180 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥𝐿𝑦)‘))
59 eqidd 2621 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎𝑥) = (𝑎𝑥))
6050, 58, 59oveq123d 6656 . . . . . . . . . 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 2635 . . . . . . . . 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 2983 . . . . . . . 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 2986 . . . . . . 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 3190 . . . . . 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 3554 . . . . 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 3554 . . . 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 4645 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
6812, 67sylib 208 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
69 df-br 4645 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
7022, 69sylib 208 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
71 ovex 6663 . . . . . . . 8 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7271rgenw 2921 . . . . . . 7 𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
73 ixpexg 7917 . . . . . . 7 (∀𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V → X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V)
7472, 73ax-mp 5 . . . . . 6 X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∈ V
7574rabex 4804 . . . . 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 6773 . . 3 (𝜑 → (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) = {𝑎X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥))})
7877eleq2d 2685 . 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 6177 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑦) = (𝐴𝑦))
8079oveq1d 6650 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)))
81 fveq1 6177 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑥) = (𝐴𝑥))
8281oveq2d 6651 . . . . . 6 (𝑎 = 𝐴 → (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))
8380, 82eqeq12d 2635 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝑎𝑥)) ↔ ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
8483ralbidv 2983 . . . 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 2986 . . 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 3357 . 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 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  csb 3526  cop 4174   class class class wbr 4644  Rel wrel 5109  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  Xcixp 7893  Basecbs 15838  Hom chom 15933  compcco 15934   Func cfunc 16495   Nat cnat 16582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-ixp 7894  df-func 16499  df-nat 16584
This theorem is referenced by:  isnat2  16589  natixp  16593  nati  16596
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