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| Mirrors > Home > MPE Home > Th. List > iswlkg | Structured version Visualization version GIF version | ||
| Description: Generalization of iswlk 29628: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
| Ref | Expression |
|---|---|
| iswlkg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| iswlkg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
| Ref | Expression |
|---|---|
| iswlkg | ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlkv 29630 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
| 2 | 3simpc 1151 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
| 5 | elex 3501 | . . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ V) | |
| 6 | ovex 7464 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) ∈ V | |
| 7 | iswlkg.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 8 | 7 | fvexi 6920 | . . . . . . 7 ⊢ 𝑉 ∈ V |
| 9 | 6, 8 | fpm 8915 | . . . . . 6 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 ∈ (𝑉 ↑pm (0...(♯‘𝐹)))) |
| 10 | 9 | elexd 3504 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 ∈ V) |
| 11 | 5, 10 | anim12i 613 | . . . 4 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
| 12 | 11 | 3adant3 1133 | . . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
| 13 | 12 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
| 14 | iswlkg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 15 | 7, 14 | iswlk 29628 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) |
| 16 | 15 | 3expib 1123 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))) |
| 17 | 4, 13, 16 | pm5.21ndd 379 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 if-wif 1063 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 {csn 4626 {cpr 4628 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑pm cpm 8867 0cc0 11155 1c1 11156 + caddc 11158 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 Vtxcvtx 29013 iEdgciedg 29014 Walkscwlks 29614 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-wlks 29617 |
| This theorem is referenced by: wlkcomp 29649 wlkl1loop 29656 upgriswlk 29659 wlkres 29688 wlkp1lem8 29698 lfgriswlk 29706 2pthnloop 29751 isclwlke 29797 0wlk 30135 1wlkd 30160 pfxwlk 35129 revwlk 35130 subgrwlk 35137 |
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