Theorem funtransport 36026

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 3210	. . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)))
4	2, 3	anim12i 613	. . . . . . . . . 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 615	. . . . . . . . 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 660	. . . . . . . 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 8003	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1st ‘𝑝) ∈ (𝔼‘𝑛))
8		xp1st 8003	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1st ‘𝑝) ∈ (𝔼‘𝑚))
9		axdimuniq 28847	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st ‘𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10		fveq2 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
11	10	riotaeqdv 7348	. . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . 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 254	. . . . . . . . . . . . 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 660	. . . . . . . . . . 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 412	. . . . . . . . . 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 607	. . . . . . . . 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 410	. . . . . . . 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 3180	. . . . . 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 235	. . . . 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 1796	. . . 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 2734	. . . . . . . 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 630	. . . . . . 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 3158	. . . . . 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 5673	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
29	28	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
30	28	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
31	29, 30	3anbi12d 1439	. . . . . . . 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 632	. . . . . . 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 3217	. . . . . 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 287	. . . . 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 2560	. . . 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 231	. . 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 7514	. 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 36025	. . 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 6540	. 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 231	1 ⊢ Fun TransportTo

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532   ≠ wne 2926  ∃wrex 3054  ⟨cop 4598   class class class wbr 5110   × cxp 5639  Fun wfun 6508  ‘cfv 6514  ℩crio 7346  {coprab 7391  1^st c1st 7969  2^nd c2nd 7970  ℕcn 12193  𝔼cee 28822   Btwn cbtwn 28823  Cgrccgr 28824  TransportToctransport 36024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-z 12537  df-uz 12801  df-fz 13476  df-ee 28825  df-transport 36025

This theorem is referenced by:  fvtransport  36027