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Theorem pthdivtx 30199

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 30193	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		trliswlk 30167	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2762	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 30084	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 13843	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)))
6		elfzo1 13772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)))
7		nnnn0 12539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
8	7	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
9	6, 8	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
10	9	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
11		fvinim0ffz 13849	. . . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = 0 → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
15	14	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
16		ffun 6709	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
17	16	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → Fun 𝑃)
18		fdm 6716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
19		fzo0ss1 13749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
20		fzossfz 13738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
21	19, 20	sstri 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
22	21	sseli 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0...(♯‘𝐹)))
23		eleq2 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(♯‘𝐹))))
24	22, 23	imbitrrid 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ dom 𝑃))
25	18, 24	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ dom 𝑃))
26	25	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → 𝐼 ∈ dom 𝑃)
27	17, 26	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
28	27	adantrl 729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
29		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → 𝐼 ∈ (1..^(♯‘𝐹)))
30		funfvima 7233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃) → (𝐼 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
31	28, 29, 30	sylc 66	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))))
32		eleq1 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘0) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
33	31, 32	syl5ibcom 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘0) → (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
34	15, 33	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
35		nnel 3073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
36	34, 35	imbitrrdi 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
37	36	necon2ad 2972	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
38	37	adantrd 497	. . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
40	39	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
41	40	com23 87	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
42	41	a1d 26	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 33	. . . . . . . . . . . 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 26	. . . . . . . . . . 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 418	. . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
48	47	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
49	48	adantl 487	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) = (𝑃‘𝐽))
52	51	ad2antrl 741	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 6745	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
56	21, 55	mpan2 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
57		df-f1 6542	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺) ↔ ((𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))))
58	57	biimpri 231	. . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
60	59	3adant3 1150	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 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 7257	. . . . . . . . . . . . . . . . 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 698	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . 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 2978	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 87	. . . . . . . . . . 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 418	. . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
72	71, 11	sylan2 605	. . . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (♯‘𝐹) → (𝑃‘𝐽) = (𝑃‘(♯‘𝐹)))
74	73	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = (♯‘𝐹) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
75	74	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
76	27	adantrl 729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
77		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → 𝐼 ∈ (1..^(♯‘𝐹)))
78	76, 77, 30	sylc 66	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))))
79		eleq1 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘(♯‘𝐹)) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
80	78, 79	syl5ibcom 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘(♯‘𝐹)) → (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
81	75, 80	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹)))))
82		nnel 3073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
83	81, 82	imbitrrdi 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
84	83	necon2ad 2972	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
85	84	adantld 496	. . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
87	86	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
88	87	com23 87	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ → ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
89	88	a1d 26	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 33	. . . . . . . . . . . 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 26	. . . . . . . . . . 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 418	. . . . . . . . . 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 1454	. . . . . . . . 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 18	. . . . . . . 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 1131	. . . . . . 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 33	. . . . . 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 1137	. . . . 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 19	. . . 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 1128	. . 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 220	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
102	101	imp 412	1 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∉ wnel 3063 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 {cpr 4589 class class class wbr 5107 ^◡ccnv 5658 dom cdm 5659 ↾ cres 5661 “ cima 5662 Fun wfun 6531 ⟶wf 6533 –1-1→wf1 6534 ‘cfv 6537 (class class class)co 7417 0cc0 11128 1c1 11129 < clt 11271 ℕcn 12261 ℕ₀cn0 12532 ...cfz 13565 ..^cfzo 13713 ♯chash 14398 Vtxcvtx 29461 Walkscwlks 30064 Trailsctrls 30160 Pathscpths 30182

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-fzo 13714 df-hash 14399 df-word 14583 df-wlks 30067 df-trls 30162 df-pths 30186

This theorem is used by: pthdadjvtx 30200 upgr4cycl4dv4e 30673 upgrimpthslem2 48832

W3C validator