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Mirrors > Home > MPE Home > Th. List > spthispth | Structured version Visualization version GIF version |
Description: A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
spthispth | ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → 𝐹(Trails‘𝐺)𝑃) | |
2 | funres11 6004 | . . . 4 ⊢ (Fun ◡𝑃 → Fun ◡(𝑃 ↾ (1..^(#‘𝐹)))) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → Fun ◡(𝑃 ↾ (1..^(#‘𝐹)))) |
4 | 1e0p1 11590 | . . . . . . . . . 10 ⊢ 1 = (0 + 1) | |
5 | 4 | oveq1i 6700 | . . . . . . . . 9 ⊢ (1..^(#‘𝐹)) = ((0 + 1)..^(#‘𝐹)) |
6 | 5 | ineq2i 3844 | . . . . . . . 8 ⊢ ({0, (#‘𝐹)} ∩ (1..^(#‘𝐹))) = ({0, (#‘𝐹)} ∩ ((0 + 1)..^(#‘𝐹))) |
7 | 0z 11426 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
8 | prinfzo0 12546 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → ({0, (#‘𝐹)} ∩ ((0 + 1)..^(#‘𝐹))) = ∅) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ ({0, (#‘𝐹)} ∩ ((0 + 1)..^(#‘𝐹))) = ∅ |
10 | 6, 9 | eqtri 2673 | . . . . . . 7 ⊢ ({0, (#‘𝐹)} ∩ (1..^(#‘𝐹))) = ∅ |
11 | 10 | imaeq2i 5499 | . . . . . 6 ⊢ (𝑃 “ ({0, (#‘𝐹)} ∩ (1..^(#‘𝐹)))) = (𝑃 “ ∅) |
12 | ima0 5516 | . . . . . 6 ⊢ (𝑃 “ ∅) = ∅ | |
13 | 11, 12 | eqtri 2673 | . . . . 5 ⊢ (𝑃 “ ({0, (#‘𝐹)} ∩ (1..^(#‘𝐹)))) = ∅ |
14 | imain 6012 | . . . . 5 ⊢ (Fun ◡𝑃 → (𝑃 “ ({0, (#‘𝐹)} ∩ (1..^(#‘𝐹)))) = ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹))))) | |
15 | 13, 14 | syl5reqr 2700 | . . . 4 ⊢ (Fun ◡𝑃 → ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) |
16 | 15 | adantl 481 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) |
17 | 1, 3, 16 | 3jca 1261 | . 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) |
18 | isspth 26676 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
19 | ispth 26675 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) | |
20 | 17, 18, 19 | 3imtr4i 281 | 1 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∩ cin 3606 ∅c0 3948 {cpr 4212 class class class wbr 4685 ◡ccnv 5142 ↾ cres 5145 “ cima 5146 Fun wfun 5920 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 ℤcz 11415 ..^cfzo 12504 #chash 13157 Trailsctrls 26643 Pathscpths 26664 SPathscspths 26665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-wlks 26551 df-trls 26645 df-pths 26668 df-spths 26669 |
This theorem is referenced by: spthiswlk 26680 spthson 26693 spthonprop 26697 isspthonpth 26701 spthonpthon 26703 usgr2trlspth 26713 usgr2pthspth 26714 wspthsnonn0vne 26882 |
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