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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 6643 | . . . 4 ⊢ (Fun ◡𝑃 → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) | |
| 3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) |
| 4 | imain 6651 | . . . . 5 ⊢ (Fun ◡𝑃 → (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹))))) | |
| 5 | 1e0p1 12775 | . . . . . . . . . 10 ⊢ 1 = (0 + 1) | |
| 6 | 5 | oveq1i 7441 | . . . . . . . . 9 ⊢ (1..^(♯‘𝐹)) = ((0 + 1)..^(♯‘𝐹)) |
| 7 | 6 | ineq2i 4217 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) |
| 8 | 0z 12624 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
| 9 | prinfzo0 13738 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅) | |
| 10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅ |
| 11 | 7, 10 | eqtri 2765 | . . . . . . 7 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ∅ |
| 12 | 11 | imaeq2i 6076 | . . . . . 6 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = (𝑃 “ ∅) |
| 13 | ima0 6095 | . . . . . 6 ⊢ (𝑃 “ ∅) = ∅ | |
| 14 | 12, 13 | eqtri 2765 | . . . . 5 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ∅ |
| 15 | 4, 14 | eqtr3di 2792 | . . . 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 1129 | . 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) |
| 18 | isspth 29742 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
| 19 | ispth 29741 | . 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 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 {cpr 4628 class class class wbr 5143 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℤcz 12613 ..^cfzo 13694 ♯chash 14369 Trailsctrls 29708 Pathscpths 29730 SPathscspths 29731 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-trls 29710 df-pths 29734 df-spths 29735 |
| This theorem is referenced by: spthiswlk 29746 isspthonpth 29769 spthonpthon 29771 usgr2trlspth 29781 usgr2pthspth 29782 pthspthcyc 29823 wspthsnonn0vne 29937 spthcycl 35134 |
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