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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pthhashvtx Structured version   Visualization version   GIF version
Theorem pthhashvtx 34118
Description: A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
Hypothesis
Ref Expression
pthhashvtx.1 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
pthhashvtx (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
Proof of Theorem pthhashvtx
StepHypRef Expression
1 hashfz0 14392 . . . 4 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ (β™―β€˜(0...((β™―β€˜πΉ) βˆ’ 1))) = (((β™―β€˜πΉ) βˆ’ 1) + 1))
2 pthiswlk 28984 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 wlkcl 28872 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
42, 3syl 17 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
5 nn0cn 12482 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
6 npcan1 11639 . . . . 5 ((β™―β€˜πΉ) ∈ β„‚ β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
74, 5, 63syl 18 . . . 4 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
81, 7sylan9eqr 2795 . . 3 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜(0...((β™―β€˜πΉ) βˆ’ 1))) = (β™―β€˜πΉ))
9 pthhashvtx.1 . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
109wlkp 28873 . . . . . . 7 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
112, 10syl 17 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
1211ffnd 6719 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
13 fzfi 13937 . . . . 5 (0...((β™―β€˜πΉ) βˆ’ 1)) ∈ Fin
14 resfnfinfin 9332 . . . . 5 ((𝑃 Fn (0...(β™―β€˜πΉ)) ∧ (0...((β™―β€˜πΉ) βˆ’ 1)) ∈ Fin) β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin)
1512, 13, 14sylancl 587 . . . 4 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin)
16 simpr 486 . . . . 5 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0)
17 fzssp1 13544 . . . . . . . 8 (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(((β™―β€˜πΉ) βˆ’ 1) + 1))
187oveq2d 7425 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (0...(β™―β€˜πΉ)))
1917, 18sseqtrid 4035 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(β™―β€˜πΉ)))
2011, 19fssresd 6759 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))):(0...((β™―β€˜πΉ) βˆ’ 1))βŸΆπ‘‰)
2120adantr 482 . . . . 5 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))):(0...((β™―β€˜πΉ) βˆ’ 1))βŸΆπ‘‰)
22 fz1ssfz0 13597 . . . . . . . . 9 (1...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...((β™―β€˜πΉ) βˆ’ 1))
2322a1i 11 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)))
2420, 23fssresd 6759 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))):(1...((β™―β€˜πΉ) βˆ’ 1))βŸΆπ‘‰)
25 ispth 28980 . . . . . . . . 9 (𝐹(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
2625simp2bi 1147 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
27 nn0z 12583 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
28 fzoval 13633 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„€ β†’ (1..^(β™―β€˜πΉ)) = (1...((β™―β€˜πΉ) βˆ’ 1)))
2927, 28syl 17 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) ∈ β„•0 β†’ (1..^(β™―β€˜πΉ)) = (1...((β™―β€˜πΉ) βˆ’ 1)))
304, 29syl 17 . . . . . . . . . . . 12 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1..^(β™―β€˜πΉ)) = (1...((β™―β€˜πΉ) βˆ’ 1)))
3130reseq2d 5982 . . . . . . . . . . 11 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))) = (𝑃 β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))))
32 resabs1 6012 . . . . . . . . . . . 12 ((1...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)) β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))) = (𝑃 β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))))
3322, 32ax-mp 5 . . . . . . . . . . 11 ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))) = (𝑃 β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1)))
3431, 33eqtr4di 2791 . . . . . . . . . 10 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))) = ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))))
3534cnveqd 5876 . . . . . . . . 9 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) = β—‘((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))))
3635funeqd 6571 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ↔ Fun β—‘((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1)))))
3726, 36mpbid 231 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ Fun β—‘((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))))
38 df-f1 6549 . . . . . . 7 (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))):(1...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉 ↔ (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))):(1...((β™―β€˜πΉ) βˆ’ 1))βŸΆπ‘‰ ∧ Fun β—‘((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1)))))
3924, 37, 38sylanbrc 584 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))):(1...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉)
4039adantr 482 . . . . 5 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β†Ύ (1...((β™―β€˜πΉ) βˆ’ 1))):(1...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉)
41 snsspr1 4818 . . . . . . . 8 {0} βŠ† {0, (β™―β€˜πΉ)}
42 imass2 6102 . . . . . . . 8 ({0} βŠ† {0, (β™―β€˜πΉ)} β†’ (𝑃 β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)}))
4341, 42ax-mp 5 . . . . . . 7 (𝑃 β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)})
44 0elfz 13598 . . . . . . . . 9 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ 0 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)))
4544snssd 4813 . . . . . . . 8 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ {0} βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)))
46 resima2 6017 . . . . . . . 8 ({0} βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)) β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) = (𝑃 β€œ {0}))
47 sseq1 4008 . . . . . . . 8 (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) = (𝑃 β€œ {0}) β†’ (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)}) ↔ (𝑃 β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)})))
4845, 46, 473syl 18 . . . . . . 7 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)}) ↔ (𝑃 β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)})))
4943, 48mpbiri 258 . . . . . 6 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)}))
50 resima2 6017 . . . . . . . . . 10 ((1...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)) β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1))) = (𝑃 β€œ (1...((β™―β€˜πΉ) βˆ’ 1))))
5122, 50ax-mp 5 . . . . . . . . 9 ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1))) = (𝑃 β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))
5230imaeq2d 6060 . . . . . . . . 9 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β€œ (1..^(β™―β€˜πΉ))) = (𝑃 β€œ (1...((β™―β€˜πΉ) βˆ’ 1))))
5351, 52eqtr4id 2792 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1))) = (𝑃 β€œ (1..^(β™―β€˜πΉ))))
5453ineq2d 4213 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))) = ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
5525simp3bi 1148 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)
5654, 55eqtrd 2773 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))) = βˆ…)
57 ssdisj 4460 . . . . . 6 ((((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) βŠ† (𝑃 β€œ {0, (β™―β€˜πΉ)}) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))) = βˆ…) β†’ (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) ∩ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))) = βˆ…)
5849, 56, 57syl2anr 598 . . . . 5 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ {0}) ∩ ((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) β€œ (1...((β™―β€˜πΉ) βˆ’ 1)))) = βˆ…)
5916, 21, 40, 58f1resfz0f1d 34103 . . . 4 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))):(0...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉)
609fvexi 6906 . . . . 5 𝑉 ∈ V
61 hashf1dmcdm 34105 . . . . 5 (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin ∧ 𝑉 ∈ V ∧ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))):(0...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉) β†’ (β™―β€˜(0...((β™―β€˜πΉ) βˆ’ 1))) ≀ (β™―β€˜π‘‰))
6260, 61mp3an2 1450 . . . 4 (((𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin ∧ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))):(0...((β™―β€˜πΉ) βˆ’ 1))–1-1→𝑉) β†’ (β™―β€˜(0...((β™―β€˜πΉ) βˆ’ 1))) ≀ (β™―β€˜π‘‰))
6315, 59, 62syl2an2r 684 . . 3 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜(0...((β™―β€˜πΉ) βˆ’ 1))) ≀ (β™―β€˜π‘‰))
648, 63eqbrtrrd 5173 . 2 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
65 0nn0m1nnn0 34102 . . . . 5 ((β™―β€˜πΉ) = 0 ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0))
6665biimpri 227 . . . 4 (((β™―β€˜πΉ) ∈ β„•0 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) = 0)
674, 66sylan 581 . . 3 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) = 0)
68 hashge0 14347 . . . 4 (𝑉 ∈ V β†’ 0 ≀ (β™―β€˜π‘‰))
6960, 68ax-mp 5 . . 3 0 ≀ (β™―β€˜π‘‰)
7067, 69eqbrtrdi 5188 . 2 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
7164, 70pm2.61dan 812 1 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  {cpr 4631   class class class wbr 5149  β—‘ccnv 5676   β†Ύ cres 5679   β€œ cima 5680  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  β„‚cc 11108  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  β„•0cn0 12472  β„€cz 12558  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Vtxcvtx 28256  Walkscwlks 28853  Trailsctrls 28947  Pathscpths 28969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wlks 28856  df-trls 28949  df-pths 28973
This theorem is referenced by:  usgrcyclgt2v  34122  acycgr1v  34140
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