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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 13600 | . . . . . . 7 β’ (0...0) = {0} | |
| 2 | 1 | eqcomi 2741 | . . . . . 6 β’ {0} = (0...0) |
| 3 | 2 | feq2i 6709 | . . . . 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 1135 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β π:(0...0)βΆ{π}) |
| 6 | snssi 4811 | . . . 4 β’ (π β π β {π} β π) | |
| 7 | 6 | 3ad2ant1 1133 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β {π} β π) |
| 8 | 5, 7 | fssd 6735 | . 2 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β π:(0...0)βΆπ) |
| 9 | breq1 5151 | . . . 4 β’ (πΉ = β β (πΉ(ClWalksβπΊ)π β β (ClWalksβπΊ)π)) | |
| 10 | 9 | 3ad2ant2 1134 | . . 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 28259 | . . . . 5 β’ (π β π β πΊ β V) |
| 13 | 11 | 0clwlk 29380 | . . . . 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 1133 | . . 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 278 | . 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 256 | 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 1087 = wceq 1541 β wcel 2106 Vcvv 3474 β wss 3948 β c0 4322 {csn 4628 class class class wbr 5148 βΆwf 6539 βcfv 6543 (class class class)co 7408 0cc0 11109 ...cfz 13483 Vtxcvtx 28253 ClWalkscclwlks 29024 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 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 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 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-wlks 28853 df-clwlks 29025 |
| This theorem is referenced by: wlkl0 29617 |
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