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| Mirrors > Home > MPE Home > Th. List > 0clwlkv | Structured version Visualization version GIF version | ||
| Description: Any vertex (more precisely, a pair of an empty set (of edges) and a singleton function to this vertex) determines a closed walk of length 0. (Contributed by AV, 11-Feb-2022.) |
| Ref | Expression |
|---|---|
| 0clwlk.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| 0clwlkv | β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β πΉ(ClWalksβπΊ)π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fz0sn 13602 | . . . . . . 7 β’ (0...0) = {0} | |
| 2 | 1 | eqcomi 2733 | . . . . . 6 β’ {0} = (0...0) |
| 3 | 2 | feq2i 6700 | . . . . 5 β’ (π:{0}βΆ{π} β π:(0...0)βΆ{π}) |
| 4 | 3 | biimpi 215 | . . . 4 β’ (π:{0}βΆ{π} β π:(0...0)βΆ{π}) |
| 5 | 4 | 3ad2ant3 1132 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β π:(0...0)βΆ{π}) |
| 6 | snssi 4804 | . . . 4 β’ (π β π β {π} β π) | |
| 7 | 6 | 3ad2ant1 1130 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β {π} β π) |
| 8 | 5, 7 | fssd 6726 | . 2 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β π:(0...0)βΆπ) |
| 9 | breq1 5142 | . . . 4 β’ (πΉ = β β (πΉ(ClWalksβπΊ)π β β (ClWalksβπΊ)π)) | |
| 10 | 9 | 3ad2ant2 1131 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β (πΉ(ClWalksβπΊ)π β β (ClWalksβπΊ)π)) |
| 11 | 0clwlk.v | . . . . . 6 β’ π = (VtxβπΊ) | |
| 12 | 11 | 1vgrex 28756 | . . . . 5 β’ (π β π β πΊ β V) |
| 13 | 11 | 0clwlk 29878 | . . . . 5 β’ (πΊ β V β (β (ClWalksβπΊ)π β π:(0...0)βΆπ)) |
| 14 | 12, 13 | syl 17 | . . . 4 β’ (π β π β (β (ClWalksβπΊ)π β π:(0...0)βΆπ)) |
| 15 | 14 | 3ad2ant1 1130 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β (β (ClWalksβπΊ)π β π:(0...0)βΆπ)) |
| 16 | 10, 15 | bitrd 279 | . 2 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β (πΉ(ClWalksβπΊ)π β π:(0...0)βΆπ)) |
| 17 | 8, 16 | mpbird 257 | 1 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β πΉ(ClWalksβπΊ)π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3466 β wss 3941 β c0 4315 {csn 4621 class class class wbr 5139 βΆwf 6530 βcfv 6534 (class class class)co 7402 0cc0 11107 ...cfz 13485 Vtxcvtx 28750 ClWalkscclwlks 29522 |
| 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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-wlks 29351 df-clwlks 29523 |
| This theorem is referenced by: wlkl0 30115 |
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