Theorem funray 36353

Description: Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funray	⊢ Fun Ray

Proof of Theorem funray

Dummy variables 𝑚 𝑎 𝑛 𝑝 𝑟 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeanv 3210	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
2		simp1 1137	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑛))
3		simp1 1137	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) → 𝑝 ∈ (𝔼‘𝑚))
4		axdimuniq 28998	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
5		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
6		rabeq 3415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝔼‘𝑛) = (𝔼‘𝑚) → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
7	5, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
8	7	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
9	8	anbi1d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ (𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
10		eqtr3 2759	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠)
11	9, 10	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
12	4, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑚))) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
13	12	an4s 661	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚))) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠))
14	13	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → 𝑟 = 𝑠)))
15	14	com3l 89	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑝 ∈ (𝔼‘𝑚)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠)))
16	2, 3, 15	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)) → ((𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠)))
17	16	imp 406	. . . . . . . . 9 ⊢ ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)) ∧ (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠))
18	17	an4s 661	. . . . . . . 8 ⊢ ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → 𝑟 = 𝑠))
19	18	com12 32	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠))
20	19	rexlimivv 3180	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
21	1, 20	sylbir 235	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
22	21	gen2 1798	. . . 4 ⊢ ∀𝑟∀𝑠((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠)
23		eqeq1 2741	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
24	23	anbi2d 631	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
25	24	rexbidv 3162	. . . . . 6 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
26	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑝 ∈ (𝔼‘𝑚)))
27	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
28	26, 27	3anbi12d 1440	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎)))
29	7	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
30	28, 29	anbi12d 633	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
31	30	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}))
32	25, 31	bitrdi 287	. . . . 5 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})))
33	32	mo4 2567	. . . 4 ⊢ (∃*𝑟∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∀𝑟∀𝑠((∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑚) ∧ 𝑎 ∈ (𝔼‘𝑚) ∧ 𝑝 ≠ 𝑎) ∧ 𝑠 = {𝑥 ∈ (𝔼‘𝑚) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})) → 𝑟 = 𝑠))
34	22, 33	mpbir 231	. . 3 ⊢ ∃*𝑟∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})
35	34	funoprab 7490	. 2 ⊢ Fun {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
36		df-ray 36351	. . 3 ⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
37	36	funeqi 6521	. 2 ⊢ (Fun Ray ↔ Fun {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
38	35, 37	mpbir 231	1 ⊢ Fun Ray

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538   ≠ wne 2933  ∃wrex 3062  {crab 3401  ⟨cop 4588   class class class wbr 5100  Fun wfun 6494  ‘cfv 6500  {coprab 7369  ℕcn 12157  𝔼cee 28972  OutsideOfcoutsideof 36332  Raycray 36348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-z 12501  df-uz 12764  df-fz 13436  df-ee 28975  df-ray 36351

This theorem is referenced by:  fvray  36354