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Mirrors > Home > MPE Home > Th. List > 1loopgrvd0 | Structured version Visualization version GIF version |
Description: The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
1loopgruspgr.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
1loopgruspgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1loopgruspgr.n | ⊢ (𝜑 → 𝑁 ∈ 𝑉) |
1loopgruspgr.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) |
1loopgrvd0.k | ⊢ (𝜑 → 𝐾 ∈ (𝑉 ∖ {𝑁})) |
Ref | Expression |
---|---|
1loopgrvd0 | ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1loopgrvd0.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑉 ∖ {𝑁})) | |
2 | 1 | eldifbd 3894 | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∈ {𝑁}) |
3 | 1loopgruspgr.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
4 | snex 5297 | . . . . . 6 ⊢ {𝑁} ∈ V | |
5 | fvsng 6919 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝑁} ∈ V) → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) | |
6 | 3, 4, 5 | sylancl 589 | . . . . 5 ⊢ (𝜑 → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) |
7 | 6 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴) ↔ 𝐾 ∈ {𝑁})) |
8 | 2, 7 | mtbird 328 | . . 3 ⊢ (𝜑 → ¬ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴)) |
9 | 1loopgruspgr.i | . . . . . . 7 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | |
10 | 9 | dmeqd 5738 | . . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝑁}〉}) |
11 | dmsnopg 6037 | . . . . . . 7 ⊢ ({𝑁} ∈ V → dom {〈𝐴, {𝑁}〉} = {𝐴}) | |
12 | 4, 11 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → dom {〈𝐴, {𝑁}〉} = {𝐴}) |
13 | 10, 12 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
14 | 9 | fveq1d 6647 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝑖)) |
15 | 14 | eleq2d 2875 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
16 | 13, 15 | rexeqbidv 3355 | . . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ ∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
17 | fveq2 6645 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → ({〈𝐴, {𝑁}〉}‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝐴)) | |
18 | 17 | eleq2d 2875 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
19 | 18 | rexsng 4574 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
20 | 3, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
21 | 16, 20 | bitrd 282 | . . 3 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
22 | 8, 21 | mtbird 328 | . 2 ⊢ (𝜑 → ¬ ∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖)) |
23 | 1 | eldifad 3893 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
24 | 1loopgruspgr.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
25 | 24 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ (Vtx‘𝐺) ↔ 𝐾 ∈ 𝑉)) |
26 | 23, 25 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (Vtx‘𝐺)) |
27 | eqid 2798 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
28 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
29 | eqid 2798 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
30 | 27, 28, 29 | vtxd0nedgb 27278 | . . 3 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (((VtxDeg‘𝐺)‘𝐾) = 0 ↔ ¬ ∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖))) |
31 | 26, 30 | syl 17 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝐾) = 0 ↔ ¬ ∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖))) |
32 | 22, 31 | mpbird 260 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∖ cdif 3878 {csn 4525 〈cop 4531 dom cdm 5519 ‘cfv 6324 0cc0 10526 Vtxcvtx 26789 iEdgciedg 26790 VtxDegcvtxdg 27255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12886 df-hash 13687 df-vtxdg 27256 |
This theorem is referenced by: 1egrvtxdg0 27301 eupth2lem3lem3 28015 |
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