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Theorem funtransport 33605

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 3320	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
2		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)))
4	2, 3	anim12i 615	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞))) → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
5	4	anim1i 617	. . . . . . . . 9 ⊢ ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
6	5	an4s 659	. . . . . . . 8 ⊢ ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
7		xp1st 7703	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1^st ‘𝑝) ∈ (𝔼‘𝑛))
8		xp1st 7703	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1^st ‘𝑝) ∈ (𝔼‘𝑚))
9		axdimuniq 26707	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
11	10	riotaeqdv 7094	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))
12	11	eqeq2d 2809	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))))
13	12	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
14		eqtr3 2820	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)
15	13, 14	syl6bir 257	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
16	9, 15	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
17	16	an4s 659	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ((1^st ‘𝑝) ∈ (𝔼‘𝑛) ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
18	17	ex 416	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((1^st ‘𝑝) ∈ (𝔼‘𝑛) ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑚)) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
19	7, 8, 18	syl2ani 609	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) → ((𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
20	19	impd 414	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘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)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
22	21	rexlimivv 3251	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
23	1, 22	sylbir 238	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
24	23	gen2 1798	. . . 4 ⊢ ∀𝑥∀𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
25		eqeq1 2802	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))))
26	25	anbi2d 631	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
27	26	rexbidv 3256	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
28	10	sqxpeqd 5551	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
29	28	eleq2d 2875	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
30	28	eleq2d 2875	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
31	29, 30	3anbi12d 1434	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ↔ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞))))
32	31, 12	anbi12d 633	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
33	32	cbvrexvw 3397	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))))
34	27, 33	syl6bb 290	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))))
35	34	mo4 2625	. . . 4 ⊢ (∃*𝑥∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∀𝑥∀𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑦 = (℩𝑟 ∈ (𝔼‘𝑚)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
36	24, 35	mpbir 234	. . 3 ⊢ ∃*𝑥∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))
37	36	funoprab 7253	. 2 ⊢ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
38		df-transport 33604	. . 3 ⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
39	38	funeqi 6345	. 2 ⊢ (Fun TransportTo ↔ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))})
40	37, 39	mpbir 234	1 ⊢ Fun TransportTo

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∃*wmo 2596 ≠ wne 2987 ∃wrex 3107 ⟨cop 4531 class class class wbr 5030 × cxp 5517 Fun wfun 6318 ‘cfv 6324 ℩crio 7092 {coprab 7136 1^st c1st 7669 2^nd c2nd 7670 ℕcn 11625 𝔼cee 26682 Btwn cbtwn 26683 Cgrccgr 26684 TransportToctransport 33603

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-z 11970 df-uz 12232 df-fz 12886 df-ee 26685 df-transport 33604

This theorem is referenced by: fvtransport 33606

W3C validator