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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 482 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → 𝐹(Trails‘𝐺)𝑃) | |
2 | funres11 6624 | . . . 4 ⊢ (Fun ◡𝑃 → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) |
4 | imain 6632 | . . . . 5 ⊢ (Fun ◡𝑃 → (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹))))) | |
5 | 1e0p1 12743 | . . . . . . . . . 10 ⊢ 1 = (0 + 1) | |
6 | 5 | oveq1i 7424 | . . . . . . . . 9 ⊢ (1..^(♯‘𝐹)) = ((0 + 1)..^(♯‘𝐹)) |
7 | 6 | ineq2i 4205 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) |
8 | 0z 12593 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
9 | prinfzo0 13697 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅ |
11 | 7, 10 | eqtri 2756 | . . . . . . 7 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ∅ |
12 | 11 | imaeq2i 6055 | . . . . . 6 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = (𝑃 “ ∅) |
13 | ima0 6074 | . . . . . 6 ⊢ (𝑃 “ ∅) = ∅ | |
14 | 12, 13 | eqtri 2756 | . . . . 5 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ∅ |
15 | 4, 14 | eqtr3di 2783 | . . . 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 1126 | . 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) |
18 | isspth 29531 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
19 | ispth 29530 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) | |
20 | 17, 18, 19 | 3imtr4i 292 | 1 ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ∅c0 4318 {cpr 4626 class class class wbr 5142 ◡ccnv 5671 ↾ cres 5674 “ cima 5675 Fun wfun 6536 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 + caddc 11135 ℤcz 12582 ..^cfzo 13653 ♯chash 14315 Trailsctrls 29497 Pathscpths 29519 SPathscspths 29520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-trls 29499 df-pths 29523 df-spths 29524 |
This theorem is referenced by: spthiswlk 29535 isspthonpth 29556 spthonpthon 29558 usgr2trlspth 29568 usgr2pthspth 29569 wspthsnonn0vne 29721 spthcycl 34733 |
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