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Mirrors > Home > MPE Home > Th. List > eupths | Structured version Visualization version GIF version |
Description: The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
eupths.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
eupths | ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6766 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) | |
2 | eupths.i | . . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼) |
4 | 3 | dmeqd 5807 | . . . . 5 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼) |
5 | foeq3 6678 | . . . . 5 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) |
7 | 6 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑔 = 𝐺) → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) |
8 | wksv 27996 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
9 | trliswlk 28074 | . . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
10 | 9 | ssopab2i 5460 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ⊆ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} |
11 | 8, 10 | ssexi 5244 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V |
12 | 11 | a1i 11 | . . 3 ⊢ (⊤ → {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V) |
13 | df-eupth 28570 | . . 3 ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}) | |
14 | 7, 12, 13 | fvmptopab 7320 | . 2 ⊢ (⊤ → (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}) |
15 | 14 | mptru 1546 | 1 ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 Vcvv 3429 class class class wbr 5073 {copab 5135 dom cdm 5584 –onto→wfo 6424 ‘cfv 6426 (class class class)co 7267 0cc0 10881 ..^cfzo 13392 ♯chash 14054 iEdgciedg 27377 Walkscwlks 27973 Trailsctrls 28067 EulerPathsceupth 28569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-fzo 13393 df-hash 14055 df-word 14228 df-wlks 27976 df-trls 28069 df-eupth 28570 |
This theorem is referenced by: iseupth 28573 |
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