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Theorem dfpth2 30314

Description: Alternate definition for a pair of classes/functions to be a path (in an undirected graph). (Contributed by AV, 4-Oct-2025.)

**Assertion**
Ref	Expression
dfpth2	⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))

Proof of Theorem dfpth2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispth 30306	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		istrl 30279	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		wlkcl 30196	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 30197	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
6		ffn 6709	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
7	6	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝑃 Fn (0...(♯‘𝐹)))
8		0elfz 13758	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
9	8	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 0 ∈ (0...(♯‘𝐹)))
10		nn0fz0 13759	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
11	10	birani 509	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
12	7, 9, 11	3jca 1146	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
13	3, 5, 12	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
14	13	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
15	2, 14	sylbi 220	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
16		fnimapr 6968	. . . . . . . . . 10 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))) → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
17	15, 16	syl 18	. . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
18	17	ineq1d 4116	. . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))))
19	18	eqeq1d 2762	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
20		disj 4402	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))))
21		fvex 6898	. . . . . . . . . 10 ⊢ (𝑃‘0) ∈ V
22		fvex 6898	. . . . . . . . . 10 ⊢ (𝑃‘(♯‘𝐹)) ∈ V
23		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘0) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
24	23	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = (𝑃‘0) → (¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
25		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘(♯‘𝐹)) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
26	25	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = (𝑃‘(♯‘𝐹)) → (¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
27	21, 22, 24, 26	ralpr 4660	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
28		df-nel 3062	. . . . . . . . . 10 ⊢ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
29	28	bicomi 227	. . . . . . . . 9 ⊢ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))
30	27, 29	bianbi 639	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
31	20, 30	bitri 278	. . . . . . 7 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
32	19, 31	bitrdi 290	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))))
33	32	anbi2d 642	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))))
34		ancom 466	. . . . . . 7 ⊢ (((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
35	34	bianass 655	. . . . . 6 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))) ↔ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
36	35	a1i 11	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))) ↔ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
37		noel 4283	. . . . . . . . . . . 12 ⊢ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅
38	37	biantru 539	. . . . . . . . . . 11 ⊢ (Fun ^◡(𝑃 ↾ ∅) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
39	38	bicomi 227	. . . . . . . . . 10 ⊢ ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅))
40	39	a1i 11	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅)))
41		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = (1..^0))
42		0le1 11839	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
43		1z 12726	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
44		0z 12704	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
45		fzon 13815	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅))
46	43, 44, 45	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅)
47	42, 46	mpbi 233	. . . . . . . . . . . . . 14 ⊢ (1..^0) = ∅
48	41, 47	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = ∅)
49	48	reseq2d 5969	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅))
50	49	cnveqd 5852	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
51	50	funeqd 6561	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
52	48	imaeq2d 6051	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ ∅))
53		ima0 6074	. . . . . . . . . . . . 13 ⊢ (𝑃 “ ∅) = ∅
54	52, 53	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = ∅)
55	54	eleq2d 2846	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ ∅))
56	55	notbid 321	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
57	51, 56	anbi12d 644	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅)))
58		oveq2 7428	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = (1...0))
59		fz10 13678	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
60	58, 59	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = ∅)
61	60	reseq2d 5969	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1...(♯‘𝐹))) = (𝑃 ↾ ∅))
62	61	cnveqd 5852	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1...(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
63	62	funeqd 6561	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
64	40, 57, 63	3bitr4d 314	. . . . . . . 8 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
65	64	a1d 26	. . . . . . 7 ⊢ ((♯‘𝐹) = 0 → (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))))))
66		df-nel 3062	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
67	66	bicomi 227	. . . . . . . . . . . 12 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))
68	67	anbi2i 635	. . . . . . . . . . 11 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
69		trliswlk 30280	. . . . . . . . . . . 12 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
70	3, 10	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
71		fzonel 13808	. . . . . . . . . . . . . . 15 ⊢ ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹))
72	71	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹)))
73	70, 72	eldifd 4050	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))))
74		1eluzge0 13007	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
75		fzoss1 13821	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
76	74, 75	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
77		fzossfz 13813	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
78	76, 77	sstrdi 3920	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
79	5, 73, 78	3jca 1146	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))))
80		resf1ext2b 7947	. . . . . . . . . . . 12 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
81	69, 79, 80	3syl 19	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
82	68, 81	bitrid 286	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
83	82	adantl 487	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
84		elnnne0 12620	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
85		elnnuz 13005	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ (♯‘𝐹) ∈ (ℤ_≥‘1))
86	84, 85	sylbb1 240	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0) → (♯‘𝐹) ∈ (ℤ_≥‘1))
87	86	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
88	69, 3, 87	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
89	88	impcom 413	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (♯‘𝐹) ∈ (ℤ_≥‘1))
90		fzisfzounsn 13915	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘1) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
91	89, 90	syl 18	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
92	91	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}) = (1...(♯‘𝐹)))
93	92	reseq2d 5969	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = (𝑃 ↾ (1...(♯‘𝐹))))
94	93	cnveqd 5852	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = ^◡(𝑃 ↾ (1...(♯‘𝐹))))
95	94	funeqd 6561	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
96	83, 95	bitrd 282	. . . . . . . 8 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
97	96	ex 418	. . . . . . 7 ⊢ ((♯‘𝐹) ≠ 0 → (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))))))
98	65, 97	pm2.61ine 3038	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
99	98	anbi1d 643	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
100	33, 36, 99	3bitrd 308	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ↔ (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
101	100	pm5.32i 585	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
102		3anass 1111	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)))
103		3anass 1111	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
104	101, 102, 103	3bitr4i 306	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
105	1, 104	bitri 278	1 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∉ wnel 3061 ∀wral 3076 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 {csn 4583 {cpr 4585 class class class wbr 5102 ^◡ccnv 5649 ↾ cres 5652 “ cima 5653 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 0cc0 11200 1c1 11201 ≤ cle 11344 ℕcn 12335 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 ..^cfzo 13788 ♯chash 14474 Vtxcvtx 29574 Walkscwlks 30177 Trailsctrls 30273 Pathscpths 30295

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-wlks 30180 df-trls 30275 df-pths 30299

This theorem is used by: pthdifv 30316

W3C validator