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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 13634 | . . . . . . 7 β’ (0...0) = {0} | |
| 2 | 1 | eqcomi 2737 | . . . . . 6 β’ {0} = (0...0) |
| 3 | 2 | feq2i 6714 | . . . . 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 1133 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β π:(0...0)βΆ{π}) |
| 6 | snssi 4812 | . . . 4 β’ (π β π β {π} β π) | |
| 7 | 6 | 3ad2ant1 1131 | . . 3 β’ ((π β π β§ πΉ = β β§ π:{0}βΆ{π}) β {π} β π) |
| 8 | 5, 7 | fssd 6740 | . 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 1132 | . . 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 28828 | . . . . 5 β’ (π β π β πΊ β V) |
| 13 | 11 | 0clwlk 29953 | . . . . 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 1131 | . . 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 1085 = wceq 1534 β wcel 2099 Vcvv 3471 β wss 3947 β c0 4323 {csn 4629 class class class wbr 5148 βΆwf 6544 βcfv 6548 (class class class)co 7420 0cc0 11139 ...cfz 13517 Vtxcvtx 28822 ClWalkscclwlks 29597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-wlks 29426 df-clwlks 29598 |
| This theorem is referenced by: wlkl0 30190 |
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