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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isupwlkg | Structured version Visualization version GIF version |
Description: Generalization of isupwlk 46124: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
Ref | Expression |
---|---|
upwlksfval.v | β’ π = (VtxβπΊ) |
upwlksfval.i | β’ πΌ = (iEdgβπΊ) |
Ref | Expression |
---|---|
isupwlkg | β’ (πΊ β π β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upwlksfval.v | . . . . 5 β’ π = (VtxβπΊ) | |
2 | upwlksfval.i | . . . . 5 β’ πΌ = (iEdgβπΊ) | |
3 | 1, 2 | upwlksfval 46123 | . . . 4 β’ (πΊ β V β (UPWalksβπΊ) = {β¨π, πβ© β£ (π β Word dom πΌ β§ π:(0...(β―βπ))βΆπ β§ βπ β (0..^(β―βπ))(πΌβ(πβπ)) = {(πβπ), (πβ(π + 1))})}) |
4 | 3 | brfvopab 7415 | . . 3 β’ (πΉ(UPWalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V)) |
5 | 4 | a1i 11 | . 2 β’ (πΊ β π β (πΉ(UPWalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V))) |
6 | elex 3462 | . . . . 5 β’ (πΊ β π β πΊ β V) | |
7 | elex 3462 | . . . . . . 7 β’ (πΉ β Word dom πΌ β πΉ β V) | |
8 | ovex 7391 | . . . . . . . . 9 β’ (0...(β―βπΉ)) β V | |
9 | 1 | fvexi 6857 | . . . . . . . . 9 β’ π β V |
10 | 8, 9 | fpm 8816 | . . . . . . . 8 β’ (π:(0...(β―βπΉ))βΆπ β π β (π βpm (0...(β―βπΉ)))) |
11 | 10 | elexd 3464 | . . . . . . 7 β’ (π:(0...(β―βπΉ))βΆπ β π β V) |
12 | 7, 11 | anim12i 614 | . . . . . 6 β’ ((πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ) β (πΉ β V β§ π β V)) |
13 | 12 | 3adant3 1133 | . . . . 5 β’ ((πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β (πΉ β V β§ π β V)) |
14 | 6, 13 | anim12i 614 | . . . 4 β’ ((πΊ β π β§ (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))})) β (πΊ β V β§ (πΉ β V β§ π β V))) |
15 | 14 | ex 414 | . . 3 β’ (πΊ β π β ((πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β (πΊ β V β§ (πΉ β V β§ π β V)))) |
16 | 3anass 1096 | . . 3 β’ ((πΊ β V β§ πΉ β V β§ π β V) β (πΊ β V β§ (πΉ β V β§ π β V))) | |
17 | 15, 16 | syl6ibr 252 | . 2 β’ (πΊ β π β ((πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β (πΊ β V β§ πΉ β V β§ π β V))) |
18 | 1, 2 | isupwlk 46124 | . . 3 β’ ((πΊ β V β§ πΉ β V β§ π β V) β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) |
19 | 18 | a1i 11 | . 2 β’ (πΊ β π β ((πΊ β V β§ πΉ β V β§ π β V) β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))})))) |
20 | 5, 17, 19 | pm5.21ndd 381 | 1 β’ (πΊ β π β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3444 {cpr 4589 class class class wbr 5106 dom cdm 5634 βΆwf 6493 βcfv 6497 (class class class)co 7358 βpm cpm 8769 0cc0 11056 1c1 11057 + caddc 11059 ...cfz 13430 ..^cfzo 13573 β―chash 14236 Word cword 14408 Vtxcvtx 27989 iEdgciedg 27990 UPWalkscupwlks 46121 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-upwlks 46122 |
This theorem is referenced by: (None) |
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