Theorem el2spthonot0 26398

Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)

Assertion

Ref	Expression
el2spthonot0	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑇,𝑏   𝑉,𝑏   𝑋,𝑏   𝑌,𝑏

Proof of Theorem el2spthonot0

Dummy variables 𝑓 𝑝 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		el2spthonot 26397	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
2		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (1st ‘𝑡) = (1st ‘⟨𝐴, 𝑏, 𝐶⟩))
3	2	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)))
4	3	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((1st ‘(1st ‘𝑡)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴))
5	2	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)))
6	5	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((2nd ‘(1st ‘𝑡)) = (𝑝‘1) ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
7		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (2nd ‘𝑡) = (2nd ‘⟨𝐴, 𝑏, 𝐶⟩))
8	7	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((2nd ‘𝑡) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
9	4, 6, 8	3anbi123d 1391	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶) ↔ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
10	9	3anbi3d 1397	. . . . . . . 8 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶)) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
11	10	2exbidv 1839	. . . . . . 7 ⊢ (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶)) ↔ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
12	11	elrab 3331	. . . . . 6 ⊢ (⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶))} ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
13	12	a1i 11	. . . . 5 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶))} ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
14		2spthonot 26393	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶))})
15	14	adantr 480	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶))})
16	15	eleq2d 2673	. . . . 5 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘𝑡)) = 𝐴 ∧ (2nd ‘(1st ‘𝑡)) = (𝑝‘1) ∧ (2nd ‘𝑡) = 𝐶))}))
17		simpr1 1060	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
18		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝‘0) = 𝐴 → (𝑝‘0) = 𝐴)
19	18	eqcoms 2618	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = (𝑝‘0) → (𝑝‘0) = 𝐴)
20	19	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘0) = 𝐴)
21	20	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘0) = 𝐴)
22	21	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘0) = 𝐴)
23		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑓) = 2 → (𝑝‘(#‘𝑓)) = (𝑝‘2))
24		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝‘2) = 𝐶 → (𝑝‘2) = 𝐶)
25	24	eqcoms 2618	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 = (𝑝‘2) → (𝑝‘2) = 𝐶)
26	25	3ad2ant3 1077	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘2) = 𝐶)
27	23, 26	sylan9eq 2664	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
28	27	3adant1 1072	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
29	28	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘(#‘𝑓)) = 𝐶)
30	17, 22, 29	3jca 1235	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶))
31		simpr2 1061	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (#‘𝑓) = 2)
32		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
33		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
34	33	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉)
35		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
36		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉)
38		oteqimp 7078	. . . . . . . . . . . . . . 15 ⊢ (⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
39	38	imp 444	. . . . . . . . . . . . . 14 ⊢ ((⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
40	32, 34, 35, 37, 39	syl13anc 1320	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
41	40	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
42		eqeq2 2621	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑝‘1) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
43	42	3ad2ant2 1076	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
44	43	3ad2ant3 1077	. . . . . . . . . . . . . 14 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
45	44	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
46	45	3anbi2d 1396	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) ↔ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
47	41, 46	mpbid 221	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
48	30, 31, 47	3jca 1235	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
49		simpr11 1138	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
50		simpr2 1061	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (#‘𝑓) = 2)
51	23	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 ↔ (𝑝‘2) = 𝐶))
52	51	3anbi3d 1397	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶)))
53		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝‘0) ∈ V
54		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑝‘0) = 𝐴 → ((𝑝‘0) ∈ V ↔ 𝐴 ∈ V))
55	53, 54	mpbii 222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝‘0) = 𝐴 → 𝐴 ∈ V)
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐴 ∈ V)
57		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑏 ∈ V
58	57	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝑏 ∈ V)
59		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝‘2) ∈ V
60		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑝‘2) = 𝐶 → ((𝑝‘2) ∈ V ↔ 𝐶 ∈ V))
61	59, 60	mpbii 222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝‘2) = 𝐶 → 𝐶 ∈ V)
62	61	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐶 ∈ V)
63		ot2ndg 7074	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ V ∧ 𝑏 ∈ V ∧ 𝐶 ∈ V) → (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏)
64	56, 58, 62, 63	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏)
65	64	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ↔ 𝑏 = (𝑝‘1)))
66		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 = (𝑝‘0) → 𝐴 = (𝑝‘0))
67	66	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑝‘0) = 𝐴 → 𝐴 = (𝑝‘0))
68	67	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐴 = (𝑝‘0))
69	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐴 = (𝑝‘0))
70		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝑏 = (𝑝‘1))
71		id 22	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐶 = (𝑝‘2) → 𝐶 = (𝑝‘2))
72	71	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝‘2) = 𝐶 → 𝐶 = (𝑝‘2))
73	72	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐶 = (𝑝‘2))
74	69, 70, 73	3jca 1235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
75	74	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
76	65, 75	sylbid 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
77	76	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) → (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
78	77	3ad2ant2 1076	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
79	78	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
80	79	3adant1 1072	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
81	80	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
82	52, 81	sylbid 229	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
83	82	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
84	83	3imp 1249	. . . . . . . . . . . 12 ⊢ (((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
85	84	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
86	49, 50, 85	3jca 1235	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
87	48, 86	impbida 873	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
88	87	adantll 746	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
89		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
90		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
91	89, 90	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ V ∧ 𝑝 ∈ V)
92		isspthonpth 26114	. . . . . . . . . . . 12 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
93	91, 92	mp3an2 1404	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
95	94	bicomd 212	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ↔ 𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝))
96	95	3anbi1d 1395	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
97	88, 96	bitrd 267	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
98	97	2exbidv 1839	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
99		eqidd 2611	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
100	33	ad2antlr 759	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉)
101		simpr 476	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
102	36	ad2antlr 759	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉)
103		otel3xp 5077	. . . . . . . 8 ⊢ ((⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
104	99, 100, 101, 102, 103	syl13anc 1320	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
105	104	biantrurd 528	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
106	98, 105	bitrd 267	. . . . 5 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
107	13, 16, 106	3bitr4rd 300	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
108	107	anbi2d 736	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
109	108	rexbidva 3031	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
110	1, 109	bitrd 267	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   SPaths cspath 26029   SPathOn cspthon 26033   2SPathOnOt c2pthonot 26384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2spthonot 26387

This theorem is referenced by:  usg2spthonot1  26417