pthdivtx - Mathbox for Alexander van der Vekens

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Theorem pthdivtx 40935

Description: The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-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		pthis1wlk 40933	. . . 4 ⊢ (𝐹(PathS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
2		wlkv 40815	. . . 4 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
3	1, 2	syl 17	. . 3 ⊢ (𝐹(PathS‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4		isPth 40929	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(PathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
5		trlis1wlk 40905	. . . . . 6 ⊢ (𝐹(TrailS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
6		eqid 2610	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	1wlkp 40821	. . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺))
8		elfz0lmr 40367	. . . . . . . . . 10 ⊢ (𝐽 ∈ (0...(#‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(#‘𝐹)) ∨ 𝐽 = (#‘𝐹)))
9		elfzo1 12385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (#‘𝐹) ∈ ℕ ∧ 𝐼 < (#‘𝐹)))
10		nnnn0 11176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝐹) ∈ ℕ → (#‘𝐹) ∈ ℕ₀)
11	10	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ∈ ℕ ∧ (#‘𝐹) ∈ ℕ ∧ 𝐼 < (#‘𝐹)) → (#‘𝐹) ∈ ℕ₀)
12	9, 11	sylbi 206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℕ₀)
13	12	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ ℕ₀)
14		fvinim0ffz 12449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (#‘𝐹) ∈ ℕ₀) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))))))
15	13, 14	sylan2 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))))))
16		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
17	16	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
18	17	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
19		ffun 5961	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
20	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → Fun 𝑃)
21		fdm 5964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(#‘𝐹)))
22		fzo0ss1 12367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
23		fzossfz 12357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
24	22, 23	sstri 3577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
25	24	sseli 3564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ (0...(#‘𝐹)))
26		eleq2 2677	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (dom 𝑃 = (0...(#‘𝐹)) → (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(#‘𝐹))))
27	25, 26	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑃 = (0...(#‘𝐹)) → (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ dom 𝑃))
28	21, 27	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ dom 𝑃))
29	28	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → 𝐼 ∈ dom 𝑃)
30	20, 29	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
31	30	adantrl 748	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
32		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → 𝐼 ∈ (1..^(#‘𝐹)))
33		funfvima 6396	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹)))))
34	31, 32, 33	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))))
35		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘𝐼) = (𝑃‘0) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
36	34, 35	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘0) → (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
37	18, 36	sylbid 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
38		nnel 2892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ (𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹))))
39	37, 38	syl6ibr 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹)))))
40	39	necon2ad 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
41	40	adantrd 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
42	15, 41	sylbid 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
43	42	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
44	43	com23 84	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
45	44	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
46	45	3imp 1249	. . . . . . . . . . . . . 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)))
47	46	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
48	47	a1d 25	. . . . . . . . . . . 12 ⊢ ((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))))
49	48	ex 449	. . . . . . . . . . 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)))))
50		fvres 6117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = (𝑃‘𝐼))
51	50	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = (𝑃‘𝐼))
52	51	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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))
53	52	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
54		fvres 6117	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (1..^(#‘𝐹)) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽) = (𝑃‘𝐽))
55	54	ad2antrl 760	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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))
56	55	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
57	53, 56	eqeq12d 2625	. . . . . . . . . . . . . . . . 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‘𝐽) ↔ ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽)))
58		fssres 5983	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (1..^(#‘𝐹)) ⊆ (0...(#‘𝐹))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺))
59	24, 58	mpan2 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺))
60		df-f1 5809	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺) ↔ ((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))))
61	60	biimpri 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
62	59, 61	sylan 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
63	62	3adant3 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
64		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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))))
65	64	ancomd 466	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))))
66		f1veqaeq 6418	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
67	63, 65, 66	syl2an2r 872	. . . . . . . . . . . . . . . . 17 ⊢ (((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))
68	57, 67	sylbid 229	. . . . . . . . . . . . . . . 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) → 𝐼 = 𝐽))
69	68	ancoms 468	. . . . . . . . . . . . . . 15 ⊢ (((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))
70	69	necon3d 2803	. . . . . . . . . . . . . 14 ⊢ (((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)))
71	70	ex 449	. . . . . . . . . . . . 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))))
72	71	com23 84	. . . . . . . . . . . 12 ⊢ ((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))))
73	72	ex 449	. . . . . . . . . . 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)))))
74	12	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ ℕ₀)
75	74, 14	sylan2 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))))))
76		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐽 = (#‘𝐹) → (𝑃‘𝐽) = (𝑃‘(#‘𝐹)))
77	76	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (#‘𝐹) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(#‘𝐹))))
78	77	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(#‘𝐹))))
79	30	adantrl 748	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
80		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → 𝐼 ∈ (1..^(#‘𝐹)))
81	79, 80, 33	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))))
82		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘𝐼) = (𝑃‘(#‘𝐹)) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
83	81, 82	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘(#‘𝐹)) → (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
84	78, 83	sylbid 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
85		nnel 2892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹))))
86	84, 85	syl6ibr 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))))
87	86	necon2ad 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
88	87	adantld 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
89	75, 88	sylbid 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
90	89	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
91	90	com23 84	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
92	91	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
93	92	3imp 1249	. . . . . . . . . . . . . 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)))
94	93	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
95	94	a1d 25	. . . . . . . . . . . 12 ⊢ ((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))))
96	95	ex 449	. . . . . . . . . . 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)))))
97	49, 73, 96	3jaoi 1383	. . . . . . . . . 10 ⊢ ((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)))))
98	8, 97	syl 17	. . . . . . . . 9 ⊢ (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)))))
99	98	3imp21 1269	. . . . . . . 8 ⊢ ((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)))
100	99	com12 32	. . . . . . 7 ⊢ ((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)))
101	100	3exp 1256	. . . . . 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)))))
102	5, 7, 101	3syl 18	. . . . 5 ⊢ (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)))))
103	102	3imp 1249	. . . 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)))
104	4, 103	syl6bi 242	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(PathS‘𝐺)𝑃 → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
105	3, 104	mpcom 37	. 2 ⊢ (𝐹(PathS‘𝐺)𝑃 → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
106	105	imp 444	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 195 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 < clt 9953 ℕcn 10897 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 1Walksc1wlks 40796 TrailSctrls 40899 PathScpths 40919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 df-trls 40901 df-pths 40923

This theorem is referenced by: pthdadjvtx 40936 upgr4cycl4dv4e 41352

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