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Theorem dfpth2 29645

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 29637	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		istrl 29610	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		wlkcl 29529	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29530	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
6		ffn 6703	. . . . . . . . . . . . . . 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 13631	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
9	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 0 ∈ (0...(♯‘𝐹)))
10		nn0fz0 13632	. . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹))))
14	3, 5, 13	syl2anc 584	. . . . . . . . . . . 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 6959	. . . . . . . . . 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 4192	. . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))))
20	19	eqeq1d 2736	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
21		disj 4423	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))))
22		fvex 6886	. . . . . . . . . 10 ⊢ (𝑃‘0) ∈ V
23		fvex 6886	. . . . . . . . . 10 ⊢ (𝑃‘(♯‘𝐹)) ∈ V
24		eleq1 2821	. . . . . . . . . . 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 2821	. . . . . . . . . . 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 4674	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (¬ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
29		df-nel 3036	. . . . . . . . . 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 627	. . . . . . . 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 630	. . . . 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 642	. . . . . 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 4311	. . . . . . . . . . . 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 7408	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = (1..^0))
43		0le1 11753	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
44		1z 12615	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
45		0z 12592	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
46		fzon 13687	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅))
47	44, 45, 46	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅)
48	43, 47	mpbi 230	. . . . . . . . . . . . . 14 ⊢ (1..^0) = ∅
49	42, 48	eqtrdi 2785	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = ∅)
50	49	reseq2d 5964	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅))
51	50	cnveqd 5853	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
52	51	funeqd 6555	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
53	49	imaeq2d 6045	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ ∅))
54		ima0 6062	. . . . . . . . . . . . 13 ⊢ (𝑃 “ ∅) = ∅
55	53, 54	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = ∅)
56	55	eleq2d 2819	. . . . . . . . . . 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 632	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 0 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ ∅) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ ∅)))
59		oveq2 7408	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = (1...0))
60		fz10 13552	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
61	59, 60	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = ∅)
62	61	reseq2d 5964	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1...(♯‘𝐹))) = (𝑃 ↾ ∅))
63	62	cnveqd 5853	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1...(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
64	63	funeqd 6555	. . . . . . . . 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 3036	. . . . . . . . . . . . 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 623	. . . . . . . . . . 11 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
70		trliswlk 29611	. . . . . . . . . . . 12 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
71	3, 10	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
72		fzonel 13680	. . . . . . . . . . . . . . 15 ⊢ ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹))
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹)))
74	71, 73	eldifd 3935	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))))
75		1eluzge0 12901	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
76		fzoss1 13693	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
77	75, 76	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
78		fzossfz 13685	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
79	77, 78	sstrdi 3969	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
80	5, 74, 79	3jca 1128	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))))
81		resf1ext2b 7926	. . . . . . . . . . . 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 12508	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
86		elnnuz 12889	. . . . . . . . . . . . . . . . . 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 13785	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}) = (1...(♯‘𝐹)))
94	93	reseq2d 5964	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = (𝑃 ↾ (1...(♯‘𝐹))))
95	94	cnveqd 5853	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = ^◡(𝑃 ↾ (1...(♯‘𝐹))))
96	95	funeqd 6555	. . . . . . . . 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 3014	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ Fun ^◡(𝑃 ↾ (1...(♯‘𝐹)))))
100	99	anbi1d 631	. . . . 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 1094	. . 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 1094	. . 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 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∉ wnel 3035 ∀wral 3050 ∖ cdif 3921 ∪ cun 3922 ∩ cin 3923 ⊆ wss 3924 ∅c0 4306 {csn 4599 {cpr 4601 class class class wbr 5117 ^◡ccnv 5651 ↾ cres 5654 “ cima 5655 Fun wfun 6522 Fn wfn 6523 ⟶wf 6524 ‘cfv 6528 (class class class)co 7400 0cc0 11122 1c1 11123 ≤ cle 11263 ℕcn 12233 ℕ₀cn0 12494 ℤcz 12581 ℤ_≥cuz 12845 ...cfz 13514 ..^cfzo 13661 ♯chash 14338 Vtxcvtx 28909 Walkscwlks 29510 Trailsctrls 29604 Pathscpths 29626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-int 4921 df-iun 4967 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7857 df-1st 7983 df-2nd 7984 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-er 8714 df-map 8837 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-card 9946 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-nn 12234 df-n0 12495 df-z 12582 df-uz 12846 df-fz 13515 df-fzo 13662 df-hash 14339 df-word 14522 df-wlks 29513 df-trls 29606 df-pths 29630

This theorem is referenced by: pthdifv 29646

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