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Theorem funtransport 34616

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)) ∧ (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 1136	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3		simp1 1136	. . . . . . . . . . 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 613	. . . . . . . . . 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 615	. . . . . . . . 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 658	. . . . . . . 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 7953	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1^st ‘𝑝) ∈ (𝔼‘𝑛))
8		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1^st ‘𝑝) ∈ (𝔼‘𝑚))
9		axdimuniq 27862	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1^st ‘𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
11	10	riotaeqdv 7314	. . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 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 253	. . . . . . . . . . . . 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 658	. . . . . . . . . . 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 413	. . . . . . . . . 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 607	. . . . . . . . 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 411	. . . . . . . 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 3196	. . . . . 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 234	. . . . 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 2740	. . . . . . . 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 629	. . . . . . 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 3175	. . . . . 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 5665	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
29	28	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
30	28	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
31	29, 30	3anbi12d 1437	. . . . . . . 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 631	. . . . . . 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 3226	. . . . . 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	bitrdi 286	. . . . 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 2564	. . . 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 230	. . 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 7478	. 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 34615	. . 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 6522	. 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 230	1 ⊢ Fun TransportTo

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃*wmo 2536 ≠ wne 2943 ∃wrex 3073 ⟨cop 4592 class class class wbr 5105 × cxp 5631 Fun wfun 6490 ‘cfv 6496 ℩crio 7312 {coprab 7358 1^st c1st 7919 2^nd c2nd 7920 ℕcn 12153 𝔼cee 27837 Btwn cbtwn 27838 Cgrccgr 27839 TransportToctransport 34614

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-z 12500 df-uz 12764 df-fz 13425 df-ee 27840 df-transport 34615

This theorem is referenced by: fvtransport 34617

W3C validator