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Theorem el2wlkonot 26396

Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

**Assertion**
Ref	Expression
el2wlkonot	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

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

Proof of Theorem el2wlkonot Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2wlkonot 26392	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶))})
2	1	eleq2d 2673	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ 𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶))}))
3		fveq2 6103	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (1^st ‘𝑡) = (1^st ‘𝑇))
4	3	fveq2d 6107	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
5	4	eqeq1d 2612	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) = 𝐴 ↔ (1^st ‘(1^st ‘𝑇)) = 𝐴))
6	3	fveq2d 6107	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
7	6	eqeq1d 2612	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ↔ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1)))
8		fveq2 6103	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
9	8	eqeq1d 2612	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((2^nd ‘𝑡) = 𝐶 ↔ (2^nd ‘𝑇) = 𝐶))
10	5, 7, 9	3anbi123d 1391	. . . . . 6 ⊢ (𝑡 = 𝑇 → (((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶) ↔ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))
11	10	3anbi3d 1397	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶)) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
12	11	2exbidv 1839	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶)) ↔ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
13	12	elrab 3331	. . 3 ⊢ (𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐶))} ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
14	2, 13	syl6bb 275	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))))
15		simpl 472	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌))
16		vex 3176	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17		vex 3176	. . . . . . . . . . 11 ⊢ 𝑝 ∈ V
18	16, 17	pm3.2i 470	. . . . . . . . . 10 ⊢ (𝑓 ∈ V ∧ 𝑝 ∈ V)
19	18	a1i 11	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑓 ∈ V ∧ 𝑝 ∈ V))
20		simpr 476	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
21		iswlkon 26062	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
22	15, 19, 20, 21	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
23	22	3anbi1d 1395	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) ↔ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
24	23	anbi2d 736	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))))
25		simpl 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
26	25	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
27	26	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → 𝐴 ∈ 𝑉)
28	27	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → 𝐴 ∈ 𝑉)
29		el2wlkonotlem 26389	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
30	29	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
31	30	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
33	32	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
34		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉)
35	34	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
36	35	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → 𝐶 ∈ 𝑉)
37	36	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → 𝐶 ∈ 𝑉)
38		el2xptp0 7103	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
39	28, 33, 37, 38	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
40		oteq2 4350	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑝‘1) = 𝑏 → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
41	40	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = (𝑝‘1) → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
42	41	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ↔ 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
43	42	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
44	43	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
45	44	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
46	45	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩)
47		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → 𝑓(𝑉 Walks 𝐸)𝑝)
48	47	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → 𝑓(𝑉 Walks 𝐸)𝑝)
49	48	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝑓(𝑉 Walks 𝐸)𝑝)
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑓(𝑉 Walks 𝐸)𝑝)
51		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (#‘𝑓) = 2)
52	51	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → (#‘𝑓) = 2)
53	52	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (#‘𝑓) = 2)
54	53	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (#‘𝑓) = 2)
55		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑝‘0) = 𝐴 ↔ 𝐴 = (𝑝‘0))
56	55	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑝‘0) = 𝐴 → 𝐴 = (𝑝‘0))
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝐴 = (𝑝‘0))
58	57	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝐴 = (𝑝‘0))
59		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑏 = (𝑝‘1))
60		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((#‘𝑓) = 2 → (𝑝‘(#‘𝑓)) = (𝑝‘2))
61	60	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 ↔ (𝑝‘2) = 𝐶))
62		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑝‘2) = 𝐶 ↔ 𝐶 = (𝑝‘2))
63	62	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑝‘2) = 𝐶 → 𝐶 = (𝑝‘2))
64	61, 63	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 → 𝐶 = (𝑝‘2)))
65	64	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → 𝐶 = (𝑝‘2)))
66	65	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → ((𝑝‘(#‘𝑓)) = 𝐶 → 𝐶 = (𝑝‘2)))
67	66	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) → 𝐶 = (𝑝‘2))
68	67	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝐶 = (𝑝‘2))
69	68	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝐶 = (𝑝‘2))
70	58, 59, 69	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
71	50, 54, 70	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
72	46, 71	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
73	72	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = (𝑝‘1) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
74	73	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝑏 = (𝑝‘1)) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
75	29, 74	rspcimedv 3284	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
76	75	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
77	76	exp41 636	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
78	77	com15 99	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
79	78	pm2.43i 50	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
80	79	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
81	80	com24 93	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝑝‘0) = 𝐴 → ((𝑝‘(#‘𝑓)) = 𝐶 → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
82	81	3imp 1249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
83	82	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
84	83	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
85	39, 84	sylbid 229	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (#‘𝑓) = 2) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
86	85	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → ((#‘𝑓) = 2 → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
87	86	com23 84	. . . . . . . . . . . . . . 15 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
88	87	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
89	88	com4t 91	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
90	89	ex 449	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶) → ((#‘𝑓) = 2 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
91	90	com14 94	. . . . . . . . . . 11 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶) → ((#‘𝑓) = 2 → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
92	91	com23 84	. . . . . . . . . 10 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
93	92	3imp 1249	. . . . . . . . 9 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
94	93	impcom 445	. . . . . . . 8 ⊢ ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
95	94	com12 32	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) → ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
96	25	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉)
97		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
98	34	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉)
99	96, 97, 98	3jca 1235	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
100		otel3xp 5077	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
101	99, 100	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
102	101	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
103	102	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
104	103	com12 32	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
105	104	adantll 746	. . . . . . . . . . 11 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
106	105	imp 444	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
107		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → 𝑓(𝑉 Walks 𝐸)𝑝)
108	107	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(𝑉 Walks 𝐸)𝑝)
109	108	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
110	109	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑓(𝑉 Walks 𝐸)𝑝)
111		eqcom 2617	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
112	111	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = (𝑝‘0) → (𝑝‘0) = 𝐴)
113	112	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘0) = 𝐴)
114	113	3ad2ant3 1077	. . . . . . . . . . . . . 14 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘0) = 𝐴)
115	114	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘0) = 𝐴)
116	115	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘0) = 𝐴)
117		eqcom 2617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
118	117	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = (𝑝‘2) → (𝑝‘2) = 𝐶)
119	118	3ad2ant3 1077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘2) = 𝐶)
120	119, 61	syl5ibr 235	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶))
121	120	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶)))
122	121	3imp 1249	. . . . . . . . . . . . . 14 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
123	122	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘(#‘𝑓)) = 𝐶)
124	123	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘(#‘𝑓)) = 𝐶)
125	110, 116, 124	3jca 1235	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶))
126		id 22	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑓) = 2 → (#‘𝑓) = 2)
127	126	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (#‘𝑓) = 2)
128	127	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (#‘𝑓) = 2)
129	128	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (#‘𝑓) = 2)
130	99	adantll 746	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
131		oteqimp 7078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝑏 ∧ (2^nd ‘𝑇) = 𝐶)))
132	130, 131	syl5 33	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝑏 ∧ (2^nd ‘𝑇) = 𝐶)))
133		eqeq2 2621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑝‘1) → ((2^nd ‘(1^st ‘𝑇)) = 𝑏 ↔ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1)))
134	133	3anbi2d 1396	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑝‘1) → (((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝑏 ∧ (2^nd ‘𝑇) = 𝐶) ↔ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))
135	134	imbi2d 329	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑝‘1) → (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝑏 ∧ (2^nd ‘𝑇) = 𝐶)) ↔ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
136	132, 135	syl5ib 233	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
137	136	3ad2ant2 1076	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
138	137	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
139	138	impcom 445	. . . . . . . . . . . 12 ⊢ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))
140	139	impcom 445	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))
141	125, 129, 140	3jca 1235	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))
142	106, 141	jca 553	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
143	142	ex 449	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))))
144	143	rexlimdva 3013	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶)))))
145	95, 144	impbid 201	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
146	24, 145	bitrd 267	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
147	146	2exbidv 1839	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓∃𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ ∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
148		19.42vv 1907	. . . 4 ⊢ (∃𝑓∃𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))))
149		rexcom4 3198	. . . . 5 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓∃𝑏 ∈ 𝑉 ∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
150		rexcom4 3198	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑝∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
151	150	exbii 1764	. . . . 5 ⊢ (∃𝑓∃𝑏 ∈ 𝑉 ∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
152	149, 151	bitr2i 264	. . . 4 ⊢ (∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
153	147, 148, 152	3bitr3g 301	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ ∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
154		19.42vv 1907	. . . 4 ⊢ (∃𝑓∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
155	154	rexbii 3023	. . 3 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
156	153, 155	syl6bb 275	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = (𝑝‘1) ∧ (2^nd ‘𝑇) = 𝐶))) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
157	14, 156	bitrd 267	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Colors of variables: wff setvar class

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 Walks cwalk 26026 WalkOn cwlkon 26030 2WalksOnOt c2wlkonot 26382

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-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-wlkon 26042 df-2wlkonot 26385

This theorem is referenced by: el2wlkonotot0 26399 el2wlksot 26407 frg2wot1 26584 frg2woteq 26587

W3C validator