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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 29342 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
3 | 2 | anbi1d 631 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) ↔ (𝑃:(0...0)⟶𝑉 ∧ Fun ◡𝑃))) |
4 | isspth 28948 | . 2 ⊢ (∅(SPaths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
5 | fz0sn 13588 | . . . . 5 ⊢ (0...0) = {0} | |
6 | 5 | feq2i 6699 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:{0}⟶𝑉) |
7 | c0ex 11195 | . . . . . 6 ⊢ 0 ∈ V | |
8 | 7 | fsn2 7121 | . . . . 5 ⊢ (𝑃:{0}⟶𝑉 ↔ ((𝑃‘0) ∈ 𝑉 ∧ 𝑃 = {〈0, (𝑃‘0)〉})) |
9 | funcnvsn 6590 | . . . . . 6 ⊢ Fun ◡{〈0, (𝑃‘0)〉} | |
10 | cnveq 5868 | . . . . . . 7 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → ◡𝑃 = ◡{〈0, (𝑃‘0)〉}) | |
11 | 10 | funeqd 6562 | . . . . . 6 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → (Fun ◡𝑃 ↔ Fun ◡{〈0, (𝑃‘0)〉})) |
12 | 9, 11 | mpbiri 258 | . . . . 5 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → Fun ◡𝑃) |
13 | 8, 12 | simplbiim 506 | . . . 4 ⊢ (𝑃:{0}⟶𝑉 → Fun ◡𝑃) |
14 | 6, 13 | sylbi 216 | . . 3 ⊢ (𝑃:(0...0)⟶𝑉 → Fun ◡𝑃) |
15 | 14 | pm4.71i 561 | . 2 ⊢ (𝑃:(0...0)⟶𝑉 ↔ (𝑃:(0...0)⟶𝑉 ∧ Fun ◡𝑃)) |
16 | 3, 4, 15 | 3bitr4g 314 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∅c0 4320 {csn 4624 〈cop 4630 class class class wbr 5144 ◡ccnv 5671 Fun wfun 6529 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 0cc0 11097 ...cfz 13471 Vtxcvtx 28223 Trailsctrls 28914 SPathscspths 28937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-fzo 13615 df-hash 14278 df-word 14452 df-wlks 28823 df-trls 28916 df-spths 28941 |
This theorem is referenced by: (None) |
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