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Mirrors > Home > MPE Home > Th. List > Mathboxes > isupwlkg | Structured version Visualization version GIF version |
Description: Generalization of isupwlk 44010: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
Ref | Expression |
---|---|
upwlksfval.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upwlksfval.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
isupwlkg | ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upwlksfval.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | upwlksfval.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | upwlksfval 44009 | . . . 4 ⊢ (𝐺 ∈ V → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
4 | 3 | brfvopab 7210 | . . 3 ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))) |
6 | elex 3512 | . . . . 5 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
7 | elex 3512 | . . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ V) | |
8 | ovex 7188 | . . . . . . . . 9 ⊢ (0...(♯‘𝐹)) ∈ V | |
9 | 1 | fvexi 6683 | . . . . . . . . 9 ⊢ 𝑉 ∈ V |
10 | 8, 9 | fpm 8438 | . . . . . . . 8 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 ∈ (𝑉 ↑pm (0...(♯‘𝐹)))) |
11 | 10 | elexd 3514 | . . . . . . 7 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 ∈ V) |
12 | 7, 11 | anim12i 614 | . . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
13 | 12 | 3adant3 1128 | . . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
14 | 6, 13 | anim12i 614 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) → (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
15 | 14 | ex 415 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
16 | 3anass 1091 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ↔ (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) | |
17 | 15, 16 | syl6ibr 254 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))) |
18 | 1, 2 | isupwlk 44010 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
19 | 18 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))) |
20 | 5, 17, 19 | pm5.21ndd 383 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 {cpr 4568 class class class wbr 5065 dom cdm 5554 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑pm cpm 8406 0cc0 10536 1c1 10537 + caddc 10539 ...cfz 12891 ..^cfzo 13032 ♯chash 13689 Word cword 13860 Vtxcvtx 26780 iEdgciedg 26781 UPWalkscupwlks 44007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-upwlks 44008 |
This theorem is referenced by: (None) |
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