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Theorem dfpth2 29797

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 29789	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		istrl 29763	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		wlkcl 29684	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29685	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
6		ffn 6669	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
7	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝑃 Fn (0...(♯‘𝐹)))
8		0elfz 13578	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
9	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 0 ∈ (0...(♯‘𝐹)))
10		nn0fz0 13579	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
11	10	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
13	7, 9, 12	3jca 1129	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
14	3, 5, 13	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
16	2, 15	sylbi 217	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
17		fnimapr 6924	. . . . . . . . . 10 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))) → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝑃 “ {0, (♯‘𝐹)}) = {(𝑃‘0), (𝑃‘(♯‘𝐹))})
19	18	ineq1d 4160	. . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))))
20	19	eqeq1d 2739	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
21		disj 4391	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))))
22		fvex 6854	. . . . . . . . . 10 ⊢ (𝑃‘0) ∈ V
23		fvex 6854	. . . . . . . . . 10 ⊢ (𝑃‘(♯‘𝐹)) ∈ V
24		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘0) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
25	24	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 = (𝑃‘0) → (¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
26		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘(♯‘𝐹)) → (𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
27	26	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 = (𝑃‘(♯‘𝐹)) → (¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
28	22, 23, 25, 27	ralpr 4645	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
29		df-nel 3038	. . . . . . . . . 10 ⊢ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
30	29	bicomi 224	. . . . . . . . 9 ⊢ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))
31	28, 30	bianbi 628	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
32	21, 31	bitri 275	. . . . . . 7 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
33	20, 32	bitrdi 287	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))))
34	33	anbi2d 631	. . . . 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)))))))
35		ancom 460	. . . . . . 7 ⊢ (((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
36	35	bianass 643	. . . . . 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)))))
37	36	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))))))
38		noel 4279	. . . . . . . . . . . 12 ⊢ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅
39	38	biantru 529	. . . . . . . . . . 11 ⊢ (Fun ^◡(𝑃 ↾ ∅) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
40	39	bicomi 224	. . . . . . . . . 10 ⊢ ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅))
41	40	a1i 11	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅) ↔ Fun ^◡(𝑃 ↾ ∅)))
42		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = (1..^0))
43		0le1 11673	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
44		1z 12557	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
45		0z 12535	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
46		fzon 13635	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅))
47	44, 45, 46	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅)
48	43, 47	mpbi 230	. . . . . . . . . . . . . 14 ⊢ (1..^0) = ∅
49	42, 48	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = ∅)
50	49	reseq2d 5945	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅))
51	50	cnveqd 5831	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
52	51	funeqd 6521	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
53	49	imaeq2d 6026	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ ∅))
54		ima0 6043	. . . . . . . . . . . . 13 ⊢ (𝑃 “ ∅) = ∅
55	53, 54	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = ∅)
56	55	eleq2d 2823	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ ∅))
57	56	notbid 318	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅))
58	52, 57	anbi12d 633	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅)))
59		oveq2 7375	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = (1...0))
60		fz10 13499	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
61	59, 60	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = ∅)
62	61	reseq2d 5945	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1...(♯‘𝐹))) = (𝑃 ↾ ∅))
63	62	cnveqd 5831	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1...(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
64	63	funeqd 6521	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
65	41, 58, 64	3bitr4d 311	. . . . . . . 8 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
66	65	a1d 25	. . . . . . 7 ⊢ ((♯‘𝐹) = 0 → (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))))))
67		df-nel 3038	. . . . . . . . . . . . 13 ⊢ ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
68	67	bicomi 224	. . . . . . . . . . . 12 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))
69	68	anbi2i 624	. . . . . . . . . . 11 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
70		trliswlk 29764	. . . . . . . . . . . 12 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
71	3, 10	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
72		fzonel 13628	. . . . . . . . . . . . . . 15 ⊢ ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹))
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹)))
74	71, 73	eldifd 3901	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))))
75		1eluzge0 12830	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
76		fzoss1 13641	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
77	75, 76	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
78		fzossfz 13633	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
79	77, 78	sstrdi 3935	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
80	5, 74, 79	3jca 1129	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))))
81		resf1ext2b 7886	. . . . . . . . . . . 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)}))))
82	70, 80, 81	3syl 18	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
83	69, 82	bitrid 283	. . . . . . . . . 10 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
84	83	adantl 481	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))))
85		elnnne0 12451	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
86		elnnuz 12828	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ (♯‘𝐹) ∈ (ℤ_≥‘1))
87	85, 86	sylbb1 237	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0) → (♯‘𝐹) ∈ (ℤ_≥‘1))
88	87	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
89	70, 3, 88	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((♯‘𝐹) ≠ 0 → (♯‘𝐹) ∈ (ℤ_≥‘1)))
90	89	impcom 407	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (♯‘𝐹) ∈ (ℤ_≥‘1))
91		fzisfzounsn 13735	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘1) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
92	90, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (1...(♯‘𝐹)) = ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}))
93	92	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}) = (1...(♯‘𝐹)))
94	93	reseq2d 5945	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = (𝑃 ↾ (1...(♯‘𝐹))))
95	94	cnveqd 5831	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = ^◡(𝑃 ↾ (1...(♯‘𝐹))))
96	95	funeqd 6521	. . . . . . . . 9 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (Fun ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
97	84, 96	bitrd 279	. . . . . . . 8 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
98	97	ex 412	. . . . . . 7 ⊢ ((♯‘𝐹) ≠ 0 → (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))))))
99	66, 98	pm2.61ine 3016	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
100	99	anbi1d 632	. . . . 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))))))
101	34, 37, 100	3bitrd 305	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ↔ (Fun ^◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
102	101	pm5.32i 574	. . 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))))))
103		3anass 1095	. . 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)))) = ∅)))
104		3anass 1095	. . 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))))))
105	102, 103, 104	3bitr4i 303	. 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)))))
106	1, 105	bitri 275	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∉ wnel 3037 ∀wral 3052 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 {cpr 4570 class class class wbr 5086 ^◡ccnv 5630 ↾ cres 5633 “ cima 5634 Fun wfun 6493 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 0cc0 11038 1c1 11039 ≤ cle 11180 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Vtxcvtx 29065 Walkscwlks 29665 Trailsctrls 29757 Pathscpths 29778

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-wlks 29668 df-trls 29759 df-pths 29782

This theorem is referenced by: pthdifv 29798

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