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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 27677 | . . 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 1096 | . . . 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 6416 | . . . . . 6 ⊢ Fun ◡∅ | |
6 | hash0 13833 | . . . . . . . . . . . 12 ⊢ (♯‘∅) = 0 | |
7 | 0le1 11254 | . . . . . . . . . . . 12 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 5061 | . . . . . . . . . . 11 ⊢ (♯‘∅) ≤ 1 |
9 | 1z 12106 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℤ | |
10 | 0z 12086 | . . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ | |
11 | 6, 10 | eqeltri 2830 | . . . . . . . . . . . 12 ⊢ (♯‘∅) ∈ ℤ |
12 | fzon 13162 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘∅) ∈ ℤ) → ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅)) | |
13 | 9, 11, 12 | mp2an 692 | . . . . . . . . . . 11 ⊢ ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅) |
14 | 8, 13 | mpbi 233 | . . . . . . . . . 10 ⊢ (1..^(♯‘∅)) = ∅ |
15 | 14 | reseq2i 5832 | . . . . . . . . 9 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = (𝑃 ↾ ∅) |
16 | res0 5839 | . . . . . . . . 9 ⊢ (𝑃 ↾ ∅) = ∅ | |
17 | 15, 16 | eqtri 2762 | . . . . . . . 8 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = ∅ |
18 | 17 | cnveqi 5727 | . . . . . . 7 ⊢ ◡(𝑃 ↾ (1..^(♯‘∅))) = ◡∅ |
19 | 18 | funeqi 6371 | . . . . . 6 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ↔ Fun ◡∅) |
20 | 5, 19 | mpbir 234 | . . . . 5 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) |
21 | 14 | imaeq2i 5911 | . . . . . . . 8 ⊢ (𝑃 “ (1..^(♯‘∅))) = (𝑃 “ ∅) |
22 | ima0 5929 | . . . . . . . 8 ⊢ (𝑃 “ ∅) = ∅ | |
23 | 21, 22 | eqtri 2762 | . . . . . . 7 ⊢ (𝑃 “ (1..^(♯‘∅))) = ∅ |
24 | 23 | ineq2i 4110 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) |
25 | in0 4290 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) = ∅ | |
26 | 24, 25 | eqtri 2762 | . . . . 5 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅ |
27 | 20, 26 | pm3.2i 474 | . . . 4 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) |
28 | 27 | biantru 533 | . . 3 ⊢ (∅(Trails‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) |
29 | 4, 28 | bitr4di 292 | . 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 28072 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
32 | 2, 29, 31 | 3bitrd 308 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∩ cin 3852 ∅c0 4221 {cpr 4528 class class class wbr 5040 ◡ccnv 5534 ↾ cres 5537 “ cima 5538 Fun wfun 6344 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 0cc0 10628 1c1 10629 ≤ cle 10767 ℤcz 12075 ...cfz 12994 ..^cfzo 13137 ♯chash 13795 Vtxcvtx 26954 Trailsctrls 27645 Pathscpths 27666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-hash 13796 df-word 13969 df-wlks 27554 df-trls 27647 df-pths 27670 |
This theorem is referenced by: 0pthon 28077 0cycl 28084 |
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