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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 3900 | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∈ {𝑁}) |
3 | 1loopgruspgr.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
4 | snex 5354 | . . . . . 6 ⊢ {𝑁} ∈ V | |
5 | fvsng 7052 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝑁} ∈ V) → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) | |
6 | 3, 4, 5 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) |
7 | 6 | eleq2d 2824 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴) ↔ 𝐾 ∈ {𝑁})) |
8 | 2, 7 | mtbird 325 | . . 3 ⊢ (𝜑 → ¬ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴)) |
9 | 1loopgruspgr.i | . . . . . . 7 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | |
10 | 9 | dmeqd 5814 | . . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝑁}〉}) |
11 | dmsnopg 6116 | . . . . . . 7 ⊢ ({𝑁} ∈ V → dom {〈𝐴, {𝑁}〉} = {𝐴}) | |
12 | 4, 11 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → dom {〈𝐴, {𝑁}〉} = {𝐴}) |
13 | 10, 12 | eqtrd 2778 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
14 | 9 | fveq1d 6776 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝑖)) |
15 | 14 | eleq2d 2824 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
16 | 13, 15 | rexeqbidv 3337 | . . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ ∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
17 | fveq2 6774 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → ({〈𝐴, {𝑁}〉}‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝐴)) | |
18 | 17 | eleq2d 2824 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
19 | 18 | rexsng 4610 | . . . . 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 278 | . . 3 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
22 | 8, 21 | mtbird 325 | . 2 ⊢ (𝜑 → ¬ ∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖)) |
23 | 1 | eldifad 3899 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
24 | 1loopgruspgr.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
25 | 24 | eleq2d 2824 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ (Vtx‘𝐺) ↔ 𝐾 ∈ 𝑉)) |
26 | 23, 25 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (Vtx‘𝐺)) |
27 | eqid 2738 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
28 | eqid 2738 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
29 | eqid 2738 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
30 | 27, 28, 29 | vtxd0nedgb 27855 | . . 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 256 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 {csn 4561 〈cop 4567 dom cdm 5589 ‘cfv 6433 0cc0 10871 Vtxcvtx 27366 iEdgciedg 27367 VtxDegcvtxdg 27832 |
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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 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 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 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 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-xadd 12849 df-fz 13240 df-hash 14045 df-vtxdg 27833 |
This theorem is referenced by: 1egrvtxdg0 27878 eupth2lem3lem3 28594 |
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