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Theorem dfpth2 29936

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 29928	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		istrl 29902	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		wlkcl 29823	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29824	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
6		ffn 6691	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
7	6	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝑃 Fn (0...(♯‘𝐹)))
8		0elfz 13639	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
9	8	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 0 ∈ (0...(♯‘𝐹)))
10		nn0fz0 13640	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
11	10	birani 507	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
12	7, 9, 11	3jca 1142	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
13	3, 5, 12	syl2anc 593	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
14	13	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
15	2, 14	sylbi 219	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
16		fnimapr 6950	. . . . . . . . . 10 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))) → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
18	17	ineq1d 4172	. . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))))
19	18	eqeq1d 2765	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
20		disj 4405	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))))
21		fvex 6880	. . . . . . . . . 10 ⊢ (𝑃‘0) ∈ V
22		fvex 6880	. . . . . . . . . 10 ⊢ (𝑃‘(♯‘𝐹)) ∈ V
23		eleq1 2851	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘0) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
24	23	notbid 320	. . . . . . . . . 10 ⊢ (𝑥 = (𝑃‘0) → (¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
25		eleq1 2851	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘(♯‘𝐹)) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
26	25	notbid 320	. . . . . . . . . 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 3063	. . . . . . . . . 10 ⊢ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
29	28	bicomi 226	. . . . . . . . 9 ⊢ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))
30	27, 29	bianbi 636	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
31	20, 30	bitri 277	. . . . . . 7 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
32	19, 31	bitrdi 289	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))))
33	32	anbi2d 639	. . . . 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 464	. . . . . . 7 ⊢ (((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
35	34	bianass 652	. . . . . 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 4291	. . . . . . . . . . . 12 ⊢ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅
38	37	biantru 537	. . . . . . . . . . 11 ⊢ (Fun ^◡(𝑃 ↾ ∅) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
39	38	bicomi 226	. . . . . . . . . 10 ⊢ ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅))
40	39	a1i 11	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅)))
41		oveq2 7404	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = (1..^0))
42		0le1 11721	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
43		1z 12611	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
44		0z 12589	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
45		fzon 13696	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅))
46	43, 44, 45	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅)
47	42, 46	mpbi 232	. . . . . . . . . . . . . 14 ⊢ (1..^0) = ∅
48	41, 47	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = ∅)
49	48	reseq2d 5965	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅))
50	49	cnveqd 5848	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
51	50	funeqd 6543	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
52	48	imaeq2d 6049	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ ∅))
53		ima0 6066	. . . . . . . . . . . . 13 ⊢ (𝑃 “ ∅) = ∅
54	52, 53	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = ∅)
55	54	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ ∅))
56	55	notbid 320	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
57	51, 56	anbi12d 641	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅)))
58		oveq2 7404	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = (1...0))
59		fz10 13560	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
60	58, 59	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = ∅)
61	60	reseq2d 5965	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1...(♯‘𝐹))) = (𝑃 ↾ ∅))
62	61	cnveqd 5848	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1...(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
63	62	funeqd 6543	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
64	40, 57, 63	3bitr4d 313	. . . . . . . 8 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
65	64	a1d 25	. . . . . . 7 ⊢ ((♯‘𝐹) = 0 → (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))))))
66		df-nel 3063	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
67	66	bicomi 226	. . . . . . . . . . . 12 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))
68	67	anbi2i 632	. . . . . . . . . . 11 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
69		trliswlk 29903	. . . . . . . . . . . 12 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
70	3, 10	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
71		fzonel 13689	. . . . . . . . . . . . . . 15 ⊢ ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹))
72	71	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹)))
73	70, 72	eldifd 3916	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))))
74		1eluzge0 12891	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
75		fzoss1 13702	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
76	74, 75	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
77		fzossfz 13694	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
78	76, 77	sstrdi 3949	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
79	5, 73, 78	3jca 1142	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))))
80		resf1ext2b 7916	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
82	68, 81	bitrid 285	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
83	82	adantl 485	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
84		elnnne0 12505	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
85		elnnuz 12889	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ (♯‘𝐹) ∈ (ℤ_≥‘1))
86	84, 85	sylbb1 239	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0) → (♯‘𝐹) ∈ (ℤ_≥‘1))
87	86	ex 416	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
88	69, 3, 87	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
89	88	impcom 411	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (♯‘𝐹) ∈ (ℤ_≥‘1))
90		fzisfzounsn 13796	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘1) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
92	91	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}) = (1...(♯‘𝐹)))
93	92	reseq2d 5965	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = (𝑃 ↾ (1...(♯‘𝐹))))
94	93	cnveqd 5848	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = ^◡(𝑃 ↾ (1...(♯‘𝐹))))
95	94	funeqd 6543	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
96	83, 95	bitrd 281	. . . . . . . 8 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
97	96	ex 416	. . . . . . 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 3041	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
99	98	anbi1d 640	. . . . 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 307	. . . 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 582	. . 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 1107	. . 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 1107	. . 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 305	. 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 277	1 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∉ wnel 3062 ∀wral 3077 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4583 {cpr 4585 class class class wbr 5101 ^◡ccnv 5647 ↾ cres 5650 “ cima 5651 Fun wfun 6515 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 0cc0 11084 1c1 11085 ≤ cle 11228 ℕcn 12220 ℕ₀cn0 12491 ℤcz 12578 ℤ_≥cuz 12849 ...cfz 13522 ..^cfzo 13669 ♯chash 14353 Vtxcvtx 29204 Walkscwlks 29804 Trailsctrls 29896 Pathscpths 29917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ifp 1075 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 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 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-n0 12492 df-z 12579 df-uz 12850 df-fz 13523 df-fzo 13670 df-hash 14354 df-word 14537 df-wlks 29807 df-trls 29898 df-pths 29921

This theorem is referenced by: pthdifv 29937

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