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Theorem pthhashvtx 33785
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 14341 . . . 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 28724 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 wlkcl 28612 . . . . . 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 12431 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
6 npcan1 11588 . . . . 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 28613 . . . . . . 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 6673 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
13 fzfi 13886 . . . . 5 (0...((β™―β€˜πΉ) βˆ’ 1)) ∈ Fin
14 resfnfinfin 9282 . . . . 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 13493 . . . . . . . 8 (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(((β™―β€˜πΉ) βˆ’ 1) + 1))
187oveq2d 7377 . . . . . . . 8 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (0...(β™―β€˜πΉ)))
1917, 18sseqtrid 4000 . . . . . . 7 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(β™―β€˜πΉ)))
2011, 19fssresd 6713 . . . . . 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 13546 . . . . . . . . 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 6713 . . . . . . 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 28720 . . . . . . . . 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 12532 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
28 fzoval 13582 . . . . . . . . . . . . . 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 5941 . . . . . . . . . . 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 5971 . . . . . . . . . . . 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 5835 . . . . . . . . 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 6527 . . . . . . . 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 6505 . . . . . . 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 4778 . . . . . . . 8 {0} βŠ† {0, (β™―β€˜πΉ)}
42 imass2 6058 . . . . . . . 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 13547 . . . . . . . . 9 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ 0 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)))
4544snssd 4773 . . . . . . . 8 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ {0} βŠ† (0...((β™―β€˜πΉ) βˆ’ 1)))
46 resima2 5976 . . . . . . . 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 3973 . . . . . . . 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 5976 . . . . . . . . . 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 6017 . . . . . . . . 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 4176 . . . . . . 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 4423 . . . . . 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 33768 . . . 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 6860 . . . . 5 𝑉 ∈ V
61 hashf1dmcdm 33772 . . . . 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 5133 . 2 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
65 0nn0m1nnn0 33767 . . . . 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 14296 . . . 4 (𝑉 ∈ V β†’ 0 ≀ (β™―β€˜π‘‰))
6960, 68ax-mp 5 . . 3 0 ≀ (β™―β€˜π‘‰)
7067, 69eqbrtrdi 5148 . 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 3447   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  {csn 4590  {cpr 4592   class class class wbr 5109  β—‘ccnv 5636   β†Ύ cres 5639   β€œ cima 5640  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889  β„‚cc 11057  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198   βˆ’ cmin 11393  β„•0cn0 12421  β„€cz 12507  ...cfz 13433  ..^cfzo 13576  β™―chash 14239  Vtxcvtx 27996  Walkscwlks 28593  Trailsctrls 28687  Pathscpths 28709
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-wlks 28596  df-trls 28689  df-pths 28713
This theorem is referenced by:  usgrcyclgt2v  33789  acycgr1v  33807
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