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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 27587 | . . 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 27191 | . 2 ⊢ (∅(SPaths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
5 | fz0sn 12861 | . . . . 5 ⊢ (0...0) = {0} | |
6 | 5 | feq2i 6381 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:{0}⟶𝑉) |
7 | c0ex 10488 | . . . . . 6 ⊢ 0 ∈ V | |
8 | 7 | fsn2 6768 | . . . . 5 ⊢ (𝑃:{0}⟶𝑉 ↔ ((𝑃‘0) ∈ 𝑉 ∧ 𝑃 = {〈0, (𝑃‘0)〉})) |
9 | funcnvsn 6281 | . . . . . 6 ⊢ Fun ◡{〈0, (𝑃‘0)〉} | |
10 | cnveq 5637 | . . . . . . 7 ⊢ (𝑃 = {〈0, (𝑃‘0)〉} → ◡𝑃 = ◡{〈0, (𝑃‘0)〉}) | |
11 | 10 | funeqd 6254 | . . . . . 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 1525 ∈ wcel 2083 ∅c0 4217 {csn 4478 〈cop 4484 class class class wbr 4968 ◡ccnv 5449 Fun wfun 6226 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 0cc0 10390 ...cfz 12746 Vtxcvtx 26468 Trailsctrls 27158 SPathscspths 27180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ifp 1056 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-er 8146 df-map 8265 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-hash 13545 df-word 13712 df-wlks 27068 df-trls 27160 df-spths 27184 |
This theorem is referenced by: (None) |
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