Theorem funtransport 35680

Description: The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funtransport	⊢ Fun TransportTo

Proof of Theorem funtransport

Dummy variables 𝑚 𝑛 𝑝 𝑞 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeanv 3217	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
2		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)))
4	2, 3	anim12i 611	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞))) → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
5	4	anim1i 613	. . . . . . . . 9 ⊢ ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
6	5	an4s 658	. . . . . . . 8 ⊢ ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
7		xp1st 8019	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1st ‘𝑝) ∈ (𝔼‘𝑛))
8		xp1st 8019	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1st ‘𝑝) ∈ (𝔼‘𝑚))
9		axdimuniq 28763	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10		fveq2 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
11	10	riotaeqdv 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))
12	11	eqeq2d 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))))
13	12	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
14		eqtr3 2751	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)
15	13, 14	biimtrrdi 253	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
16	9, 15	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
17	16	an4s 658	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ((1st ‘𝑝) ∈ (𝔼‘𝑛) ∧ (1st ‘𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
18	17	ex 411	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((1st ‘𝑝) ∈ (𝔼‘𝑛) ∧ (1st ‘𝑝) ∈ (𝔼‘𝑚)) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
19	7, 8, 18	syl2ani 605	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
20	19	impd 409	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
21	6, 20	syl5 34	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
22	21	rexlimivv 3190	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
23	1, 22	sylbir 234	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
24	23	gen2 1790	. . . 4 ⊢ ∀𝑥∀𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
25		eqeq1 2729	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))))
26	25	anbi2d 628	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
27	26	rexbidv 3169	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
28	10	sqxpeqd 5705	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
29	28	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
30	28	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
31	29, 30	3anbi12d 1433	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ↔ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞))))
32	31, 12	anbi12d 630	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
33	32	cbvrexvw 3226	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))))
34	27, 33	bitrdi 286	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
35	34	mo4 2554	. . . 4 ⊢ (∃*𝑥∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∀𝑥∀𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
36	24, 35	mpbir 230	. . 3 ⊢ ∃*𝑥∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))
37	36	funoprab 7536	. 2 ⊢ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
38		df-transport 35679	. . 3 ⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
39	38	funeqi 6569	. 2 ⊢ (Fun TransportTo ↔ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))})
40	37, 39	mpbir 230	1 ⊢ Fun TransportTo

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∃*wmo 2526   ≠ wne 2930  ∃wrex 3060  ⟨cop 4631   class class class wbr 5144   × cxp 5671  Fun wfun 6537  ‘cfv 6543  ℩crio 7368  {coprab 7414  1^st c1st 7985  2^nd c2nd 7986  ℕcn 12237  𝔼cee 28738   Btwn cbtwn 28739  Cgrccgr 28740  TransportToctransport 35678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-z 12584  df-uz 12848  df-fz 13512  df-ee 28741  df-transport 35679

This theorem is referenced by:  fvtransport  35681