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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 47068: 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 47067 | . . . 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 7461 | . . 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 3487 | . . . . 5 β’ (πΊ β π β πΊ β V) | |
7 | elex 3487 | . . . . . . 7 β’ (πΉ β Word dom πΌ β πΉ β V) | |
8 | ovex 7437 | . . . . . . . . 9 β’ (0...(β―βπΉ)) β V | |
9 | 1 | fvexi 6898 | . . . . . . . . 9 β’ π β V |
10 | 8, 9 | fpm 8868 | . . . . . . . 8 β’ (π:(0...(β―βπΉ))βΆπ β π β (π βpm (0...(β―βπΉ)))) |
11 | 10 | elexd 3489 | . . . . . . 7 β’ (π:(0...(β―βπΉ))βΆπ β π β V) |
12 | 7, 11 | anim12i 612 | . . . . . 6 β’ ((πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ) β (πΉ β V β§ π β V)) |
13 | 12 | 3adant3 1129 | . . . . 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 612 | . . . 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 412 | . . 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 1092 | . . 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 | imbitrrdi 251 | . 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 47068 | . . 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 379 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 Vcvv 3468 {cpr 4625 class class class wbr 5141 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 βpm cpm 8820 0cc0 11109 1c1 11110 + caddc 11112 ...cfz 13487 ..^cfzo 13630 β―chash 14292 Word cword 14467 Vtxcvtx 28759 iEdgciedg 28760 UPWalkscupwlks 47065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-upwlks 47066 |
This theorem is referenced by: (None) |
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