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Theorem funtransport 36032
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.
StepHypRef Expression
1 reeanv 3229 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2 simp1 1137 . . . . . . . . . . 11 ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3 simp1 1137 . . . . . . . . . . 11 ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) → 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)))
42, 3anim12i 613 . . . . . . . . . 10 (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))) → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
54anim1i 615 . . . . . . . . 9 ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
65an4s 660 . . . . . . . 8 ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
7 xp1st 8046 . . . . . . . . . 10 (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1st𝑝) ∈ (𝔼‘𝑛))
8 xp1st 8046 . . . . . . . . . 10 (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1st𝑝) ∈ (𝔼‘𝑚))
9 axdimuniq 28928 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
1110riotaeqdv 7389 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))
1211eqeq2d 2748 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
1312anbi2d 630 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
14 eqtr3 2763 . . . . . . . . . . . . . 14 ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)
1513, 14biimtrrdi 254 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
169, 15syl 17 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
1716an4s 660 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ((1st𝑝) ∈ (𝔼‘𝑛) ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
1817ex 412 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((1st𝑝) ∈ (𝔼‘𝑛) ∧ (1st𝑝) ∈ (𝔼‘𝑚)) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
197, 8, 18syl2ani 607 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
2019impd 410 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
216, 20syl5 34 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
2221rexlimivv 3201 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
231, 22sylbir 235 . . . . 5 ((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
2423gen2 1796 . . . 4 𝑥𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
25 eqeq1 2741 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
2625anbi2d 630 . . . . . . 7 (𝑥 = 𝑦 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2726rexbidv 3179 . . . . . 6 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2810sqxpeqd 5717 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
2928eleq2d 2827 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
3028eleq2d 2827 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
3129, 303anbi12d 1439 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ↔ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))))
3231, 12anbi12d 632 . . . . . . 7 (𝑛 = 𝑚 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
3332cbvrexvw 3238 . . . . . 6 (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
3427, 33bitrdi 287 . . . . 5 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
3534mo4 2566 . . . 4 (∃*𝑥𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∀𝑥𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
3624, 35mpbir 231 . . 3 ∃*𝑥𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))
3736funoprab 7555 . 2 Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))}
38 df-transport 36031 . . 3 TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))}
3938funeqi 6587 . 2 (Fun TransportTo ↔ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))})
4037, 39mpbir 231 1 Fun TransportTo
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wne 2940  wrex 3070  cop 4632   class class class wbr 5143   × cxp 5683  Fun wfun 6555  cfv 6561  crio 7387  {coprab 7432  1st c1st 8012  2nd c2nd 8013  cn 12266  𝔼cee 28903   Btwn cbtwn 28904  Cgrccgr 28905  TransportToctransport 36030
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-z 12614  df-uz 12879  df-fz 13548  df-ee 28906  df-transport 36031
This theorem is referenced by:  fvtransport  36033
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