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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pthhashvtx Structured version   Visualization version   GIF version
Theorem pthhashvtx 34416
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 14396 . . . 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 29251 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 wlkcl 29139 . . . . . 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 12486 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
6 npcan1 11643 . . . . 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 2792 . . 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 29140 . . . . . . 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 6717 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
13 fzfi 13941 . . . . 5 (0...((β™―β€˜πΉ) βˆ’ 1)) ∈ Fin
14 resfnfinfin 9334 . . . . 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 584 . . . 4 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝑃 β†Ύ (0...((β™―β€˜πΉ) βˆ’ 1))) ∈ Fin)
16 simpr 483 . . . . 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 13548 . . . . . . . 8 (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(((β™―β€˜πΉ) βˆ’ 1) + 1))
187oveq2d 7427 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (0...(β™―β€˜πΉ)))
1917, 18sseqtrid 4033 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(β™―β€˜πΉ)))
2011, 19fssresd 6757 . . . . . 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 479 . . . . 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 13601 . . . . . . . . 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 6757 . . . . . . 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 29247 . . . . . . . . 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 1144 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))
27 nn0z 12587 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
28 fzoval 13637 . . . . . . . . . . . . . 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 5980 . . . . . . . . . . 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 6010 . . . . . . . . . . . 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 2788 . . . . . . . . . 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 5874 . . . . . . . . 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 6569 . . . . . . . 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 6547 . . . . . . 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 581 . . . . . 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 479 . . . . 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 4816 . . . . . . . 8 {0} βŠ† {0, (β™―β€˜πΉ)}
42 imass2 6100 . . . . . . . 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 13602 . . . . . . . . 9 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ 0 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)))
4544snssd 4811 . . . . . . . 8 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ {0} βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)))
46 resima2 6015 . . . . . . . 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 4006 . . . . . . . 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 6015 . . . . . . . . . 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 6058 . . . . . . . . 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 2789 . . . . . . . 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 4211 . . . . . . 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 1145 . . . . . . 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 2770 . . . . . 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 4458 . . . . . 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 595 . . . . 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 34401 . . . 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 6904 . . . . 5 𝑉 ∈ V
61 hashf1dmcdm 34403 . . . . 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 1447 . . . 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 681 . . 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 5171 . 2 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
65 0nn0m1nnn0 34400 . . . . 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 578 . . 3 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ Β¬ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) = 0)
68 hashge0 14351 . . . 4 (𝑉 ∈ V β†’ 0 ≀ (β™―β€˜π‘‰))
6960, 68ax-mp 5 . . 3 0 ≀ (β™―β€˜π‘‰)
7067, 69eqbrtrdi 5186 . 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 809 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  {cpr 4629   class class class wbr 5147  β—‘ccnv 5674   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  Fincfn 8941  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   ≀ cle 11253   βˆ’ cmin 11448  β„•0cn0 12476  β„€cz 12562  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Vtxcvtx 28523  Walkscwlks 29120  Trailsctrls 29214  Pathscpths 29236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-wlks 29123  df-trls 29216  df-pths 29240
This theorem is referenced by:  usgrcyclgt2v  34420  acycgr1v  34438
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