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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 27504 | . . 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 1091 | . . . 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 6420 | . . . . . 6 ⊢ Fun ◡∅ | |
6 | hash0 13729 | . . . . . . . . . . . 12 ⊢ (♯‘∅) = 0 | |
7 | 0le1 11163 | . . . . . . . . . . . 12 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 5087 | . . . . . . . . . . 11 ⊢ (♯‘∅) ≤ 1 |
9 | 1z 12013 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℤ | |
10 | 0z 11993 | . . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ | |
11 | 6, 10 | eqeltri 2909 | . . . . . . . . . . . 12 ⊢ (♯‘∅) ∈ ℤ |
12 | fzon 13059 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘∅) ∈ ℤ) → ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅)) | |
13 | 9, 11, 12 | mp2an 690 | . . . . . . . . . . 11 ⊢ ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅) |
14 | 8, 13 | mpbi 232 | . . . . . . . . . 10 ⊢ (1..^(♯‘∅)) = ∅ |
15 | 14 | reseq2i 5850 | . . . . . . . . 9 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = (𝑃 ↾ ∅) |
16 | res0 5857 | . . . . . . . . 9 ⊢ (𝑃 ↾ ∅) = ∅ | |
17 | 15, 16 | eqtri 2844 | . . . . . . . 8 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = ∅ |
18 | 17 | cnveqi 5745 | . . . . . . 7 ⊢ ◡(𝑃 ↾ (1..^(♯‘∅))) = ◡∅ |
19 | 18 | funeqi 6376 | . . . . . 6 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ↔ Fun ◡∅) |
20 | 5, 19 | mpbir 233 | . . . . 5 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) |
21 | 14 | imaeq2i 5927 | . . . . . . . 8 ⊢ (𝑃 “ (1..^(♯‘∅))) = (𝑃 “ ∅) |
22 | ima0 5945 | . . . . . . . 8 ⊢ (𝑃 “ ∅) = ∅ | |
23 | 21, 22 | eqtri 2844 | . . . . . . 7 ⊢ (𝑃 “ (1..^(♯‘∅))) = ∅ |
24 | 23 | ineq2i 4186 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) |
25 | in0 4345 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) = ∅ | |
26 | 24, 25 | eqtri 2844 | . . . . 5 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅ |
27 | 20, 26 | pm3.2i 473 | . . . 4 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) |
28 | 27 | biantru 532 | . . 3 ⊢ (∅(Trails‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) |
29 | 4, 28 | syl6bbr 291 | . 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 27901 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
32 | 2, 29, 31 | 3bitrd 307 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ∅c0 4291 {cpr 4569 class class class wbr 5066 ◡ccnv 5554 ↾ cres 5557 “ cima 5558 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 ≤ cle 10676 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Vtxcvtx 26781 Trailsctrls 27472 Pathscpths 27493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-wlks 27381 df-trls 27474 df-pths 27497 |
This theorem is referenced by: 0pthon 27906 0cycl 27913 |
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