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Theorem dfpth2 29716

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 29708	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		istrl 29681	. . . . . . . . . . 11 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		wlkcl 29600	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkp 29601	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
6		ffn 6711	. . . . . . . . . . . . . . 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 13646	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
9	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 0 ∈ (0...(♯‘𝐹)))
10		nn0fz0 13647	. . . . . . . . . . . . . . . 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 6967	. . . . . . . . . 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 4199	. . . . . . . 8 ⊢ (𝐹(Trails‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))))
20	19	eqeq1d 2738	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
21		disj 4430	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘(♯‘𝐹))} ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ {(𝑃‘0), (𝑃‘(♯‘𝐹))} ¬ 𝑥 ∈ (𝑃 “ (1..^(♯‘𝐹))))
22		fvex 6894	. . . . . . . . . 10 ⊢ (𝑃‘0) ∈ V
23		fvex 6894	. . . . . . . . . 10 ⊢ (𝑃‘(♯‘𝐹)) ∈ V
24		eleq1 2823	. . . . . . . . . . 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 2823	. . . . . . . . . . 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 4681	. . . . . . . . 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 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 4318	. . . . . . . . . . . 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 7418	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = (1..^0))
43		0le1 11765	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
44		1z 12627	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
45		0z 12604	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
46		fzon 13702	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1..^(♯‘𝐹)) = ∅)
50	49	reseq2d 5971	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ ∅))
51	50	cnveqd 5860	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
52	51	funeqd 6563	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡(𝑃 ↾ ∅)))
53	49	imaeq2d 6052	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ ∅))
54		ima0 6069	. . . . . . . . . . . . 13 ⊢ (𝑃 “ ∅) = ∅
55	53, 54	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (𝑃 “ (1..^(♯‘𝐹))) = ∅)
56	55	eleq2d 2821	. . . . . . . . . . 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 7418	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = (1...0))
60		fz10 13567	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
61	59, 60	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 0 → (1...(♯‘𝐹)) = ∅)
62	61	reseq2d 5971	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 0 → (𝑃 ↾ (1...(♯‘𝐹))) = (𝑃 ↾ ∅))
63	62	cnveqd 5860	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 0 → ^◡(𝑃 ↾ (1...(♯‘𝐹))) = ^◡(𝑃 ↾ ∅))
64	63	funeqd 6563	. . . . . . . . 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 623	. . . . . . . . . . 11 ⊢ ((Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ¬ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))) ↔ (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
70		trliswlk 29682	. . . . . . . . . . . 12 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
71	3, 10	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
72		fzonel 13695	. . . . . . . . . . . . . . 15 ⊢ ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹))
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → ¬ (♯‘𝐹) ∈ (1..^(♯‘𝐹)))
74	71, 73	eldifd 3942	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ((0...(♯‘𝐹)) ∖ (1..^(♯‘𝐹))))
75		1eluzge0 12913	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
76		fzoss1 13708	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
77	75, 76	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
78		fzossfz 13700	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
79	77, 78	sstrdi 3976	. . . . . . . . . . . . 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 7936	. . . . . . . . . . . 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 12520	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
86		elnnuz 12901	. . . . . . . . . . . . . . . . . 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 13800	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)}) = (1...(♯‘𝐹)))
94	93	reseq2d 5971	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = (𝑃 ↾ (1...(♯‘𝐹))))
95	94	cnveqd 5860	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ≠ 0 ∧ 𝐹(Trails‘𝐺)𝑃) → ^◡(𝑃 ↾ ((1..^(♯‘𝐹)) ∪ {(♯‘𝐹)})) = ^◡(𝑃 ↾ (1...(♯‘𝐹))))
96	95	funeqd 6563	. . . . . . . . 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 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 1540 ∈ wcel 2109 ≠ wne 2933 ∉ wnel 3037 ∀wral 3052 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 {csn 4606 {cpr 4608 class class class wbr 5124 ^◡ccnv 5658 ↾ cres 5661 “ cima 5662 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 0cc0 11134 1c1 11135 ≤ cle 11275 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ..^cfzo 13676 ♯chash 14353 Vtxcvtx 28980 Walkscwlks 29581 Trailsctrls 29675 Pathscpths 29697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

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 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-hash 14354 df-word 14537 df-wlks 29584 df-trls 29677 df-pths 29701

This theorem is referenced by: pthdifv 29717

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