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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pthhashvtx Structured version   Visualization version   GIF version
Theorem pthhashvtx 34113
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 14391 . . . 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 28981 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 wlkcl 28869 . . . . . 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 12481 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
6 npcan1 11638 . . . . 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 2794 . . 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 28870 . . . . . . 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 6718 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
13 fzfi 13936 . . . . 5 (0...((β™―β€˜πΉ) βˆ’ 1)) ∈ Fin
14 resfnfinfin 9331 . . . . 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 586 . . . 4 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin)
16 simpr 485 . . . . 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 13543 . . . . . . . 8 (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(((β™―β€˜πΉ) βˆ’ 1) + 1))
187oveq2d 7424 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (0...(β™―β€˜πΉ)))
1917, 18sseqtrid 4034 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(β™―β€˜πΉ)))
2011, 19fssresd 6758 . . . . . 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 481 . . . . 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 13596 . . . . . . . . 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 6758 . . . . . . 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 28977 . . . . . . . . 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 1146 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
27 nn0z 12582 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
28 fzoval 13632 . . . . . . . . . . . . . 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 5981 . . . . . . . . . . 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 6011 . . . . . . . . . . . 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 2790 . . . . . . . . . 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 5875 . . . . . . . . 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 6570 . . . . . . . 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 6548 . . . . . . 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 583 . . . . . 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 481 . . . . 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 4817 . . . . . . . 8 {0} βŠ† {0, (β™―β€˜πΉ)}
42 imass2 6101 . . . . . . . 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 13597 . . . . . . . . 9 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ 0 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)))
4544snssd 4812 . . . . . . . 8 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ {0} βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)))
46 resima2 6016 . . . . . . . 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 4007 . . . . . . . 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 257 . . . . . 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 6016 . . . . . . . . . 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 6059 . . . . . . . . 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 2791 . . . . . . . 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 4212 . . . . . . 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 1147 . . . . . . 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 2772 . . . . . 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 4459 . . . . . 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 597 . . . . 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 34098 . . . 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 6905 . . . . 5 𝑉 ∈ V
61 hashf1dmcdm 34100 . . . . 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 1449 . . . 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 683 . . 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 5172 . 2 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
65 0nn0m1nnn0 34097 . . . . 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 580 . . 3 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) = 0)
68 hashge0 14346 . . . 4 (𝑉 ∈ V β†’ 0 ≀ (β™―β€˜π‘‰))
6960, 68ax-mp 5 . . 3 0 ≀ (β™―β€˜π‘‰)
7067, 69eqbrtrdi 5187 . 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 811 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  {cpr 4630   class class class wbr 5148  β—‘ccnv 5675   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7408  Fincfn 8938  β„‚cc 11107  0cc0 11109  1c1 11110   + caddc 11112   ≀ cle 11248   βˆ’ cmin 11443  β„•0cn0 12471  β„€cz 12557  ...cfz 13483  ..^cfzo 13626  β™―chash 14289  Vtxcvtx 28253  Walkscwlks 28850  Trailsctrls 28944  Pathscpths 28966
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-oadd 8469  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-dju 9895  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-wlks 28853  df-trls 28946  df-pths 28970
This theorem is referenced by:  usgrcyclgt2v  34117  acycgr1v  34135
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