pthdivtx - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pthdivtx		Structured version Visualization version GIF version

Theorem pthdivtx 30345

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 30339	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		trliswlk 30313	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2763	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 30230	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 13948	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)))
6		elfzo1 13877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)))
7		nnnn0 12643	. . . . . . . . . . . . . . . . . . . . . 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 13954	. . . . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
19		fzo0ss1 13854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
20		fzossfz 13843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
21	19, 20	sstri 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
22	21	sseli 3909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0...(♯‘𝐹)))
23		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7262	. . . . . . . . . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . . . . . . 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 3075	. . . . . . . . . . . . . . . . . . . . 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 2974	. . . . . . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . . . . 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 6566	. . . . . . . . . . . . . . . . . . . 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 7286	. . . . . . . . . . . . . . . . 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 2980	. . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (♯‘𝐹) → (𝑃‘𝐽) = (𝑃‘(♯‘𝐹)))
74	73	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . . . . . . 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 3075	. . . . . . . . . . . . . . . . . . . . 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 2974	. . . . . . . . . . . . . . . . . . 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 1455	. . . . . . . . 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 2147 ≠ wne 2959 ∉ wnel 3065 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 {cpr 4590 class class class wbr 5106 ^◡ccnv 5655 dom cdm 5656 ↾ cres 5658 “ cima 5659 Fun wfun 6555 ⟶wf 6557 ⟶1-1wf1 6558 ‘cfv 6561 (class class class)co 7446 0cc0 11230 1c1 11231 < clt 11373 ℕcn 12365 ℕ₀cn0 12636 ...cfz 13669 ..^cfzo 13818 ♯chash 14504 Vtxcvtx 29607 Walkscwlks 30210 Trailsctrls 30306 Pathscpths 30328

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-map 8880 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-n0 12637 df-z 12724 df-uz 12996 df-fz 13670 df-fzo 13819 df-hash 14505 df-word 14689 df-wlks 30213 df-trls 30308 df-pths 30332

This theorem is used by: pthdadjvtx 30346 upgr4cycl4dv4e 30819 upgrimpthslem2 49221

W3C validator