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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 26619 | . . 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 1042 | . . . 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 5955 | . . . . . 6 ⊢ Fun ◡∅ | |
6 | hash0 13158 | . . . . . . . . . . . 12 ⊢ (#‘∅) = 0 | |
7 | 0le1 10551 | . . . . . . . . . . . 12 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 4674 | . . . . . . . . . . 11 ⊢ (#‘∅) ≤ 1 |
9 | 1z 11407 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℤ | |
10 | 0z 11388 | . . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ | |
11 | 6, 10 | eqeltri 2697 | . . . . . . . . . . . 12 ⊢ (#‘∅) ∈ ℤ |
12 | fzon 12489 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (#‘∅) ∈ ℤ) → ((#‘∅) ≤ 1 ↔ (1..^(#‘∅)) = ∅)) | |
13 | 9, 11, 12 | mp2an 708 | . . . . . . . . . . 11 ⊢ ((#‘∅) ≤ 1 ↔ (1..^(#‘∅)) = ∅) |
14 | 8, 13 | mpbi 220 | . . . . . . . . . 10 ⊢ (1..^(#‘∅)) = ∅ |
15 | 14 | reseq2i 5393 | . . . . . . . . 9 ⊢ (𝑃 ↾ (1..^(#‘∅))) = (𝑃 ↾ ∅) |
16 | res0 5400 | . . . . . . . . 9 ⊢ (𝑃 ↾ ∅) = ∅ | |
17 | 15, 16 | eqtri 2644 | . . . . . . . 8 ⊢ (𝑃 ↾ (1..^(#‘∅))) = ∅ |
18 | 17 | cnveqi 5297 | . . . . . . 7 ⊢ ◡(𝑃 ↾ (1..^(#‘∅))) = ◡∅ |
19 | 18 | funeqi 5909 | . . . . . 6 ⊢ (Fun ◡(𝑃 ↾ (1..^(#‘∅))) ↔ Fun ◡∅) |
20 | 5, 19 | mpbir 221 | . . . . 5 ⊢ Fun ◡(𝑃 ↾ (1..^(#‘∅))) |
21 | 14 | imaeq2i 5464 | . . . . . . . 8 ⊢ (𝑃 “ (1..^(#‘∅))) = (𝑃 “ ∅) |
22 | ima0 5481 | . . . . . . . 8 ⊢ (𝑃 “ ∅) = ∅ | |
23 | 21, 22 | eqtri 2644 | . . . . . . 7 ⊢ (𝑃 “ (1..^(#‘∅))) = ∅ |
24 | 23 | ineq2i 3811 | . . . . . 6 ⊢ ((𝑃 “ {0, (#‘∅)}) ∩ (𝑃 “ (1..^(#‘∅)))) = ((𝑃 “ {0, (#‘∅)}) ∩ ∅) |
25 | in0 3968 | . . . . . 6 ⊢ ((𝑃 “ {0, (#‘∅)}) ∩ ∅) = ∅ | |
26 | 24, 25 | eqtri 2644 | . . . . 5 ⊢ ((𝑃 “ {0, (#‘∅)}) ∩ (𝑃 “ (1..^(#‘∅)))) = ∅ |
27 | 20, 26 | pm3.2i 471 | . . . 4 ⊢ (Fun ◡(𝑃 ↾ (1..^(#‘∅))) ∧ ((𝑃 “ {0, (#‘∅)}) ∩ (𝑃 “ (1..^(#‘∅)))) = ∅) |
28 | 27 | biantru 526 | . . 3 ⊢ (∅(Trails‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(#‘∅))) ∧ ((𝑃 “ {0, (#‘∅)}) ∩ (𝑃 “ (1..^(#‘∅)))) = ∅))) |
29 | 4, 28 | syl6bbr 278 | . 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 26983 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
32 | 2, 29, 31 | 3bitrd 294 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ∅c0 3915 {cpr 4179 class class class wbr 4653 ◡ccnv 5113 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 ≤ cle 10075 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Vtxcvtx 25874 Trailsctrls 26587 Pathscpths 26608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-trls 26589 df-pths 26612 |
This theorem is referenced by: 0pthon 26988 0cycl 26994 |
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