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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 6665 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) | |
2 | eupths.i | . . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼) |
4 | 3 | dmeqd 5769 | . . . . 5 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼) |
5 | foeq3 6583 | . . . . 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 484 | . . 3 ⊢ ((⊤ ∧ 𝑔 = 𝐺) → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) |
8 | wksv 27395 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
9 | trliswlk 27473 | . . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
10 | 9 | ssopab2i 5430 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ⊆ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} |
11 | 8, 10 | ssexi 5219 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V |
12 | 11 | a1i 11 | . . 3 ⊢ (⊤ → {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V) |
13 | df-eupth 27971 | . . 3 ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}) | |
14 | 7, 12, 13 | fvmptopab 7203 | . 2 ⊢ (⊤ → (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}) |
15 | 14 | mptru 1540 | 1 ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 {copab 5121 dom cdm 5550 –onto→wfo 6348 ‘cfv 6350 (class class class)co 7150 0cc0 10531 ..^cfzo 13027 ♯chash 13684 iEdgciedg 26776 Walkscwlks 27372 Trailsctrls 27466 EulerPathsceupth 27970 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 df-trls 27468 df-eupth 27971 |
This theorem is referenced by: iseupth 27974 |
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