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 27770

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 27764	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		trliswlk 27739	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2736	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 27658	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 13322	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)))
6		elfzo1 13257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)))
7		nnnn0 12062	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
8	7	3ad2ant2 1136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
9	6, 8	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℕ₀)
10	9	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
11		fvinim0ffz 13326	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 6695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . 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 6526	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
17	16	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → Fun 𝑃)
18		fdm 6532	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
19		fzo0ss1 13237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
20		fzossfz 13226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
21	19, 20	sstri 3896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
22	21	sseli 3883	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0...(♯‘𝐹)))
23		eleq2 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(♯‘𝐹))))
24	22, 23	syl5ibr 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → 𝐼 ∈ dom 𝑃)
27	17, 26	jca 515	. . . . . . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → 𝐼 ∈ (1..^(♯‘𝐹)))
30		funfvima 7024	. . . . . . . . . . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . . . . . . . . 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 3045	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(♯‘𝐹))))
36	34, 35	syl6ibr 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
37	36	necon2ad 2947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
38	37	adantrd 495	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 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 1113	. . . . . . . . . . . . 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 416	. . . . . . . . . 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 6714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
48	47	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (1..^(♯‘𝐹)) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐼) = (𝑃‘𝐼))
49	48	adantl 485	. . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 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 6714	. . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . 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 6563	. . . . . . . . . . . . . . . . . . . 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 6363	. . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
60	59	3adant3 1134	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 7047	. . . . . . . . . . . . . . . . 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 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 462	. . . . . . . . . . . . . 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 2953	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 416	. . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℕ₀)
72	71, 11	sylan2 596	. . . . . . . . . . . . . . . . . 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 6695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (♯‘𝐹) → (𝑃‘𝐽) = (𝑃‘(♯‘𝐹)))
74	73	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . . . . . . . . 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 3045	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ (𝑃 “ (1..^(♯‘𝐹))))
83	81, 82	syl6ibr 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
84	83	necon2ad 2947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) ∉ (𝑃 “ (1..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
85	84	adantld 494	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 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 1113	. . . . . . . . . . . . 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 416	. . . . . . . . . 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 1429	. . . . . . . . 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 1116	. . . . . . 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 1121	. . . . 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 1113	. . 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 410	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 209 ∧ wa 399 ∨ w3o 1088 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∉ wnel 3036 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 {cpr 4529 class class class wbr 5039 ^◡ccnv 5535 dom cdm 5536 ↾ cres 5538 “ cima 5539 Fun wfun 6352 ⟶wf 6354 –1-1→wf1 6355 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 < clt 10832 ℕcn 11795 ℕ₀cn0 12055 ...cfz 13060 ..^cfzo 13203 ♯chash 13861 Vtxcvtx 27041 Walkscwlks 27638 Trailsctrls 27732 Pathscpths 27753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-wlks 27641 df-trls 27734 df-pths 27757

This theorem is referenced by: pthdadjvtx 27771 upgr4cycl4dv4e 28222

W3C validator