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Mirrors > Home > MPE Home > Th. List > 0spth | Structured version Visualization version GIF version |
Description: A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
0pth.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0spth | ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pth.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | 0trl 27828 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
3 | 2 | anbi1d 629 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) ↔ (𝑃:(0...0)⟶𝑉 ∧ Fun ◡𝑃))) |
4 | isspth 27432 | . 2 ⊢ (∅(SPaths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
5 | fz0sn 12995 | . . . . 5 ⊢ (0...0) = {0} | |
6 | 5 | feq2i 6499 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:{0}⟶𝑉) |
7 | c0ex 10623 | . . . . . 6 ⊢ 0 ∈ V | |
8 | 7 | fsn2 6890 | . . . . 5 ⊢ (𝑃:{0}⟶𝑉 ↔ ((𝑃‘0) ∈ 𝑉 ∧ 𝑃 = {〈0, (𝑃‘0)〉})) |
9 | funcnvsn 6397 | . . . . . 6 ⊢ Fun ◡{〈0, (𝑃‘0)〉} | |
10 | cnveq 5737 | . . . . . . 7 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → ◡𝑃 = ◡{〈0, (𝑃‘0)〉}) | |
11 | 10 | funeqd 6370 | . . . . . 6 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → (Fun ◡𝑃 ↔ Fun ◡{〈0, (𝑃‘0)〉})) |
12 | 9, 11 | mpbiri 259 | . . . . 5 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → Fun ◡𝑃) |
13 | 8, 12 | simplbiim 505 | . . . 4 ⊢ (𝑃:{0}⟶𝑉 → Fun ◡𝑃) |
14 | 6, 13 | sylbi 218 | . . 3 ⊢ (𝑃:(0...0)⟶𝑉 → Fun ◡𝑃) |
15 | 14 | pm4.71i 560 | . 2 ⊢ (𝑃:(0...0)⟶𝑉 ↔ (𝑃:(0...0)⟶𝑉 ∧ Fun ◡𝑃)) |
16 | 3, 4, 15 | 3bitr4g 315 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ...cfz 12880 Vtxcvtx 26708 Trailsctrls 27399 SPathscspths 27421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-wlks 27308 df-trls 27401 df-spths 27425 |
This theorem is referenced by: (None) |
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