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Theorem pthdivtx 29663

Description: The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)

**Assertion**
Ref	Expression
pthdivtx	⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))

**Proof of Theorem pthdivtx**
Step	Hyp	Ref	Expression
1		ispth 29657	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		trliswlk 29631	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2730	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 29550	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 13749	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)))
6		elfzo1 13679	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)))
7		nnnn0 12455	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
8	7	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
9	6, 8	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
10	9	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
11		fvinim0ffz 13753	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ₀) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
12	10, 11	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
13		fveq2 6860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = 0 → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
15	14	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
16		ffun 6693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
17	16	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → Fun 𝑃)
18		fdm 6699	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
19		fzo0ss1 13656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
20		fzossfz 13645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
21	19, 20	sstri 3958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
22	21	sseli 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0...(♯‘𝐹)))
23		eleq2 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(♯‘𝐹))))
24	22, 23	imbitrrid 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ dom 𝑃))
25	18, 24	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ dom 𝑃))
26	25	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → 𝐼 ∈ dom 𝑃)
27	17, 26	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
28	27	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
29		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → 𝐼 ∈ (1..^(♯‘𝐹)))
30		funfvima 7206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
31	28, 29, 30	sylc 65	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))))
32		eleq1 2817	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘0) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
33	31, 32	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘0) → (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
34	15, 33	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
35		nnel 3040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
36	34, 35	imbitrrdi 252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
37	36	necon2ad 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
38	37	adantrd 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
39	12, 38	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
40	39	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
41	40	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
42	41	a1d 25	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
43	42	3imp 1110	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
44	43	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
45	44	a1d 25	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
46	45	ex 412	. . . . . . . . . 10 ⊢ (𝐽 = 0 → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
47		fvres 6879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
48	47	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
49	48	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
50	49	eqcomd 2736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) = ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼))
51		fvres 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) = (𝑃‘𝐽))
52	51	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) = (𝑃‘𝐽))
53	52	eqcomd 2736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐽) = ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽))
54	50, 53	eqeq12d 2746	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽)))
55		fssres 6728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
56	21, 55	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
57		df-f1 6518	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺) ↔ ((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))))
58	57	biimpri 228	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
59	56, 58	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
60	59	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
61		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))))
62	61	ancomd 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (1..^(♯‘𝐹))))
63		f1veqaeq 7233	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (1..^(♯‘𝐹)))) → (((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) → 𝐼 = 𝐽))
64	60, 62, 63	syl2an2r 685	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) → 𝐼 = 𝐽))
65	54, 64	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ (((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → 𝐼 = 𝐽))
66	65	ancoms 458	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) ∧ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) → ((𝑃‘𝐼) = (𝑃‘𝐽) → 𝐼 = 𝐽))
67	66	necon3d 2947	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) ∧ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) → (𝐼 ≠ 𝐽 → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
68	67	ex 412	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝐼 ≠ 𝐽 → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
69	68	com23 86	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
70	69	ex 412	. . . . . . . . . 10 ⊢ (𝐽 ∈ (1..^(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
71	9	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
72	71, 11	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))))))
73		fveq2 6860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (♯‘𝐹) → (𝑃‘𝐽) = (𝑃‘(♯‘𝐹)))
74	73	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = (♯‘𝐹) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
75	74	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
76	27	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
77		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → 𝐼 ∈ (1..^(♯‘𝐹)))
78	76, 77, 30	sylc 65	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))))
79		eleq1 2817	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘(♯‘𝐹)) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
80	78, 79	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘(♯‘𝐹)) → (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
81	75, 80	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
82		nnel 3040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
83	81, 82	imbitrrdi 252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
84	83	necon2ad 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
85	84	adantld 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ∧ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
86	72, 85	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
87	86	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
88	87	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
89	88	a1d 25	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
90	89	3imp 1110	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
91	90	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
92	91	a1d 25	. . . . . . . . . . 11 ⊢ ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
93	92	ex 412	. . . . . . . . . 10 ⊢ (𝐽 = (♯‘𝐹) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
94	46, 70, 93	3jaoi 1430	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
95	5, 94	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
96	95	3imp21 1113	. . . . . . 7 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
97	96	com12 32	. . . . . 6 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
98	97	3exp 1119	. . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
99	2, 4, 98	3syl 18	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
100	99	3imp 1110	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
101	1, 100	sylbi 217	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
102	101	imp 406	1 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∉ wnel 3030 ∩ cin 3915 ⊆ wss 3916 ∅c0 4298 {cpr 4593 class class class wbr 5109 ^◡ccnv 5639 dom cdm 5640 ↾ cres 5642 “ cima 5643 Fun wfun 6507 ⟶wf 6509 –1-1→wf1 6510 ‘cfv 6513 (class class class)co 7389 0cc0 11074 1c1 11075 < clt 11214 ℕcn 12187 ℕ₀cn0 12448 ...cfz 13474 ..^cfzo 13621 ♯chash 14301 Vtxcvtx 28929 Walkscwlks 29530 Trailsctrls 29624 Pathscpths 29646

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-n0 12449 df-z 12536 df-uz 12800 df-fz 13475 df-fzo 13622 df-hash 14302 df-word 14485 df-wlks 29533 df-trls 29626 df-pths 29650

This theorem is referenced by: pthdadjvtx 29664 upgr4cycl4dv4e 30120 upgrimpthslem2 47898

W3C validator