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Mirrors > Home > MPE Home > Th. List > 0pth | Structured version Visualization version GIF version |
Description: A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
0pth.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0pth | ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispth 28909 | . . 3 ⊢ (∅(Paths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) |
3 | 3anass 1095 | . . . 4 ⊢ ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅)))) |
5 | funcnv0 6604 | . . . . . 6 ⊢ Fun ◡∅ | |
6 | hash0 14311 | . . . . . . . . . . . 12 ⊢ (♯‘∅) = 0 | |
7 | 0le1 11721 | . . . . . . . . . . . 12 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 5163 | . . . . . . . . . . 11 ⊢ (♯‘∅) ≤ 1 |
9 | 1z 12576 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℤ | |
10 | 0z 12553 | . . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ | |
11 | 6, 10 | eqeltri 2829 | . . . . . . . . . . . 12 ⊢ (♯‘∅) ∈ ℤ |
12 | fzon 13637 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘∅) ∈ ℤ) → ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅)) | |
13 | 9, 11, 12 | mp2an 690 | . . . . . . . . . . 11 ⊢ ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅) |
14 | 8, 13 | mpbi 229 | . . . . . . . . . 10 ⊢ (1..^(♯‘∅)) = ∅ |
15 | 14 | reseq2i 5971 | . . . . . . . . 9 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = (𝑃 ↾ ∅) |
16 | res0 5978 | . . . . . . . . 9 ⊢ (𝑃 ↾ ∅) = ∅ | |
17 | 15, 16 | eqtri 2760 | . . . . . . . 8 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = ∅ |
18 | 17 | cnveqi 5867 | . . . . . . 7 ⊢ ◡(𝑃 ↾ (1..^(♯‘∅))) = ◡∅ |
19 | 18 | funeqi 6559 | . . . . . 6 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ↔ Fun ◡∅) |
20 | 5, 19 | mpbir 230 | . . . . 5 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) |
21 | 14 | imaeq2i 6048 | . . . . . . . 8 ⊢ (𝑃 “ (1..^(♯‘∅))) = (𝑃 “ ∅) |
22 | ima0 6066 | . . . . . . . 8 ⊢ (𝑃 “ ∅) = ∅ | |
23 | 21, 22 | eqtri 2760 | . . . . . . 7 ⊢ (𝑃 “ (1..^(♯‘∅))) = ∅ |
24 | 23 | ineq2i 4206 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) |
25 | in0 4388 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) = ∅ | |
26 | 24, 25 | eqtri 2760 | . . . . 5 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅ |
27 | 20, 26 | pm3.2i 471 | . . . 4 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) |
28 | 27 | biantru 530 | . . 3 ⊢ (∅(Trails‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) |
29 | 4, 28 | bitr4di 288 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ ∅(Trails‘𝐺)𝑃)) |
30 | 0pth.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
31 | 30 | 0trl 29304 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
32 | 2, 29, 31 | 3bitrd 304 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3944 ∅c0 4319 {cpr 4625 class class class wbr 5142 ◡ccnv 5669 ↾ cres 5672 “ cima 5673 Fun wfun 6527 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 0cc0 11094 1c1 11095 ≤ cle 11233 ℤcz 12542 ...cfz 13468 ..^cfzo 13611 ♯chash 14274 Vtxcvtx 28185 Trailsctrls 28876 Pathscpths 28898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-pm 8808 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-fzo 13612 df-hash 14275 df-word 14449 df-wlks 28785 df-trls 28878 df-pths 28902 |
This theorem is referenced by: 0pthon 29309 0cycl 29316 |
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