![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clwwlknon2x | Structured version Visualization version GIF version |
Description: The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon2.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon2x.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon2x.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
clwwlknon2x | ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknon2.c | . . 3 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
2 | 1 | clwwlknon2 27887 | . 2 ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} |
3 | clwwlkn2 27829 | . . . . 5 ⊢ (𝑤 ∈ (2 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) | |
4 | 3 | anbi1i 626 | . . . 4 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) |
5 | 3anan12 1093 | . . . . . 6 ⊢ (((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))) | |
6 | 5 | anbi1i 626 | . . . . 5 ⊢ ((((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋)) |
7 | anass 472 | . . . . . 6 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))) | |
8 | clwwlknon2x.v | . . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺) | |
9 | 8 | eqcomi 2807 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) = 𝑉 |
10 | 9 | wrdeqi 13880 | . . . . . . . 8 ⊢ Word (Vtx‘𝐺) = Word 𝑉 |
11 | 10 | eleq2i 2881 | . . . . . . 7 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉) |
12 | df-3an 1086 | . . . . . . . 8 ⊢ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋)) | |
13 | clwwlknon2x.e | . . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺) | |
14 | 13 | eleq2i 2881 | . . . . . . . . . 10 ⊢ ({(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) |
15 | 14 | anbi2i 625 | . . . . . . . . 9 ⊢ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) |
16 | 15 | anbi1i 626 | . . . . . . . 8 ⊢ ((((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) |
17 | 12, 16 | bitr2i 279 | . . . . . . 7 ⊢ ((((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)) |
18 | 11, 17 | anbi12i 629 | . . . . . 6 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))) |
19 | 7, 18 | bitri 278 | . . . . 5 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))) |
20 | 6, 19 | bitri 278 | . . . 4 ⊢ ((((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))) |
21 | 4, 20 | bitri 278 | . . 3 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))) |
22 | 21 | rabbia2 3424 | . 2 ⊢ {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)} |
23 | 2, 22 | eqtri 2821 | 1 ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 {cpr 4527 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 2c2 11680 ♯chash 13686 Word cword 13857 Vtxcvtx 26789 Edgcedg 26840 ClWWalksN cclwwlkn 27809 ClWWalksNOncclwwlknon 27872 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-lsw 13906 df-clwwlk 27767 df-clwwlkn 27810 df-clwwlknon 27873 |
This theorem is referenced by: s2elclwwlknon2 27889 clwwlknon2num 27890 |
Copyright terms: Public domain | W3C validator |