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Theorem pthdivtx 29812

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 29806	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
2		trliswlk 29781	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2737	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 29702	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 13711	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(♯‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(♯‘𝐹)) ∨ 𝐽 = (♯‘𝐹)))
6		elfzo1 13640	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝐼 < (♯‘𝐹)))
7		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
8	7	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . 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 13717	. . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = 0 → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
15	14	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
16		ffun 6673	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6679	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
19		fzo0ss1 13617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
20		fzossfz 13606	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
21	19, 20	sstri 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
22	21	sseli 3931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0...(♯‘𝐹)))
23		eleq2 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . 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 3047	. . . . . . . . . . . . . . . . . . . . 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 2948	. . . . . . . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (1..^(♯‘𝐹)) → ((𝑃 ↾ (1..^(♯‘𝐹)))‘𝐽) = (𝑃‘𝐽))
52	51	ad2antrl 729	. . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . 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 6708	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
56	21, 55	mpan2 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))⟶(Vtx‘𝐺))
57		df-f1 6505	. . . . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹)))) → (𝑃 ↾ (1..^(♯‘𝐹))):(1..^(♯‘𝐹))–1-1→(Vtx‘𝐺))
60	59	3adant3 1133	. . . . . . . . . . . . . . . . 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 7212	. . . . . . . . . . . . . . . . 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 686	. . . . . . . . . . . . . . . 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 2954	. . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (♯‘𝐹) → (𝑃‘𝐽) = (𝑃‘(♯‘𝐹)))
74	73	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = (♯‘𝐹) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
75	74	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (♯‘𝐹) ∧ 𝐼 ∈ (1..^(♯‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(♯‘𝐹))))
76	27	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . 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 3047	. . . . . . . . . . . . . . . . . . . . 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 2948	. . . . . . . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . 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 1431	. . . . . . . . 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 1114	. . . . . . 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 1120	. . . . 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 1111	. . 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 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∉ wnel 3037 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 {cpr 4584 class class class wbr 5100 ^◡ccnv 5631 dom cdm 5632 ↾ cres 5634 “ cima 5635 Fun wfun 6494 ⟶wf 6496 –1-1→wf1 6497 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 < clt 11178 ℕcn 12157 ℕ₀cn0 12413 ...cfz 13435 ..^cfzo 13582 ♯chash 14265 Vtxcvtx 29081 Walkscwlks 29682 Trailsctrls 29774 Pathscpths 29795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-wlks 29685 df-trls 29776 df-pths 29799

This theorem is referenced by: pthdadjvtx 29813 upgr4cycl4dv4e 30272 upgrimpthslem2 48262

W3C validator