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Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkl1loop Structured version   Visualization version   GIF version
Theorem wlkl1loop 29162
Description: A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
Assertion
Ref Expression
wlkl1loop (((Fun (iEdgβ€˜πΊ) ∧ 𝐹(Walksβ€˜πΊ)𝑃) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))
Proof of Theorem wlkl1loop
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 wlkv 29136 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 simp3l 1199 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ Fun (iEdgβ€˜πΊ))
3 simp2 1135 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ 𝐹(Walksβ€˜πΊ)𝑃)
4 c0ex 11212 . . . . . . . . . . . . 13 0 ∈ V
54snid 4663 . . . . . . . . . . . 12 0 ∈ {0}
6 oveq2 7419 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) = 1 β†’ (0..^(β™―β€˜πΉ)) = (0..^1))
7 fzo01 13718 . . . . . . . . . . . . 13 (0..^1) = {0}
86, 7eqtrdi 2786 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 1 β†’ (0..^(β™―β€˜πΉ)) = {0})
95, 8eleqtrrid 2838 . . . . . . . . . . 11 ((β™―β€˜πΉ) = 1 β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
109ad2antrl 724 . . . . . . . . . 10 ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
11103ad2ant3 1133 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
12 eqid 2730 . . . . . . . . . 10 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
1312iedginwlk 29161 . . . . . . . . 9 ((Fun (iEdgβ€˜πΊ) ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ 0 ∈ (0..^(β™―β€˜πΉ))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) ∈ ran (iEdgβ€˜πΊ))
142, 3, 11, 13syl3anc 1369 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) ∈ ran (iEdgβ€˜πΊ))
15 eqid 2730 . . . . . . . . . . 11 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1615, 12iswlkg 29137 . . . . . . . . . 10 (𝐺 ∈ V β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))))))
178raleqdv 3323 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = 1 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ βˆ€π‘˜ ∈ {0}if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))))
18 oveq1 7418 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 0 β†’ (π‘˜ + 1) = (0 + 1))
19 0p1e1 12338 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
2018, 19eqtrdi 2786 . . . . . . . . . . . . . . . . 17 (π‘˜ = 0 β†’ (π‘˜ + 1) = 1)
21 wkslem2 29132 . . . . . . . . . . . . . . . . 17 ((π‘˜ = 0 ∧ (π‘˜ + 1) = 1) β†’ (if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))))
2220, 21mpdan 683 . . . . . . . . . . . . . . . 16 (π‘˜ = 0 β†’ (if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))))
234, 22ralsn 4684 . . . . . . . . . . . . . . 15 (βˆ€π‘˜ ∈ {0}if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
2417, 23bitrdi 286 . . . . . . . . . . . . . 14 ((β™―β€˜πΉ) = 1 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))))
2524ad2antrl 724 . . . . . . . . . . . . 13 ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))))
26 ifptru 1072 . . . . . . . . . . . . . . . . 17 ((π‘ƒβ€˜0) = (π‘ƒβ€˜1) β†’ (if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))) ↔ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}))
2726biimpa 475 . . . . . . . . . . . . . . . 16 (((π‘ƒβ€˜0) = (π‘ƒβ€˜1) ∧ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)})
2827eqcomd 2736 . . . . . . . . . . . . . . 15 (((π‘ƒβ€˜0) = (π‘ƒβ€˜1) ∧ if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))
2928ex 411 . . . . . . . . . . . . . 14 ((π‘ƒβ€˜0) = (π‘ƒβ€˜1) β†’ (if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
3029ad2antll 725 . . . . . . . . . . . . 13 ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ (if-((π‘ƒβ€˜0) = (π‘ƒβ€˜1), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0)}, {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
3125, 30sylbid 239 . . . . . . . . . . . 12 ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
3231com12 32 . . . . . . . . . . 11 (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
33323ad2ant3 1133 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))) β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0))))
3416, 33syl6bi 252 . . . . . . . . 9 (𝐺 ∈ V β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))))
35343imp 1109 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ {(π‘ƒβ€˜0)} = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)))
36 edgval 28576 . . . . . . . . 9 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
3736a1i 11 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ))
3814, 35, 373eltr4d 2846 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹(Walksβ€˜πΊ)𝑃 ∧ (Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))
39383exp 1117 . . . . . 6 (𝐺 ∈ V β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))))
40393ad2ant1 1131 . . . . 5 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))))
411, 40mpcom 38 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((Fun (iEdgβ€˜πΊ) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ)))
4241expd 414 . . 3 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (Fun (iEdgβ€˜πΊ) β†’ (((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))))
4342impcom 406 . 2 ((Fun (iEdgβ€˜πΊ) ∧ 𝐹(Walksβ€˜πΊ)𝑃) β†’ (((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1)) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ)))
4443imp 405 1 (((Fun (iEdgβ€˜πΊ) ∧ 𝐹(Walksβ€˜πΊ)𝑃) ∧ ((β™―β€˜πΉ) = 1 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜1))) β†’ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394  if-wif 1059   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βŠ† wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  dom cdm 5675  ran crn 5676  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468  Vtxcvtx 28523  iEdgciedg 28524  Edgcedg 28574  Walkscwlks 29120
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-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  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-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-edg 28575  df-wlks 29123
This theorem is referenced by:  clwlkl1loop  29307  loop1cycl  34426
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