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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 6645 | . . . 4 ⊢ (Fun ◡𝑃 → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) |
4 | imain 6653 | . . . . 5 ⊢ (Fun ◡𝑃 → (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹))))) | |
5 | 1e0p1 12773 | . . . . . . . . . 10 ⊢ 1 = (0 + 1) | |
6 | 5 | oveq1i 7441 | . . . . . . . . 9 ⊢ (1..^(♯‘𝐹)) = ((0 + 1)..^(♯‘𝐹)) |
7 | 6 | ineq2i 4225 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) |
8 | 0z 12622 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
9 | prinfzo0 13735 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ ({0, (♯‘𝐹)} ∩ ((0 + 1)..^(♯‘𝐹))) = ∅ |
11 | 7, 10 | eqtri 2763 | . . . . . . 7 ⊢ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹))) = ∅ |
12 | 11 | imaeq2i 6078 | . . . . . 6 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = (𝑃 “ ∅) |
13 | ima0 6097 | . . . . . 6 ⊢ (𝑃 “ ∅) = ∅ | |
14 | 12, 13 | eqtri 2763 | . . . . 5 ⊢ (𝑃 “ ({0, (♯‘𝐹)} ∩ (1..^(♯‘𝐹)))) = ∅ |
15 | 4, 14 | eqtr3di 2790 | . . . 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 1127 | . 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃) → (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) |
18 | isspth 29757 | . 2 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | |
19 | ispth 29756 | . 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 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 {cpr 4633 class class class wbr 5148 ◡ccnv 5688 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 ℤcz 12611 ..^cfzo 13691 ♯chash 14366 Trailsctrls 29723 Pathscpths 29745 SPathscspths 29746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-trls 29725 df-pths 29749 df-spths 29750 |
This theorem is referenced by: spthiswlk 29761 isspthonpth 29782 spthonpthon 29784 usgr2trlspth 29794 usgr2pthspth 29795 wspthsnonn0vne 29947 spthcycl 35114 |
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