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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3553 | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∈ {𝑁}) |
3 | 1loopgruspgr.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
4 | snex 4835 | . . . . . 6 ⊢ {𝑁} ∈ V | |
5 | fvsng 6352 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝑁} ∈ V) → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) | |
6 | 3, 4, 5 | sylancl 693 | . . . . 5 ⊢ (𝜑 → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) |
7 | 6 | eleq2d 2673 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴) ↔ 𝐾 ∈ {𝑁})) |
8 | 2, 7 | mtbird 314 | . . 3 ⊢ (𝜑 → ¬ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴)) |
9 | 1loopgruspgr.i | . . . . . . 7 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | |
10 | 9 | dmeqd 5248 | . . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝑁}〉}) |
11 | dmsnopg 5524 | . . . . . . 7 ⊢ ({𝑁} ∈ V → dom {〈𝐴, {𝑁}〉} = {𝐴}) | |
12 | 4, 11 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → dom {〈𝐴, {𝑁}〉} = {𝐴}) |
13 | 10, 12 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
14 | 9 | fveq1d 6105 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝑖)) |
15 | 14 | eleq2d 2673 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
16 | 13, 15 | rexeqbidv 3130 | . . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ ∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
17 | fveq2 6103 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → ({〈𝐴, {𝑁}〉}‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝐴)) | |
18 | 17 | eleq2d 2673 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
19 | 18 | rexsng 4166 | . . . . 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 267 | . . 3 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
22 | 8, 21 | mtbird 314 | . 2 ⊢ (𝜑 → ¬ ∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖)) |
23 | 1 | eldifad 3552 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
24 | 1loopgruspgr.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
25 | 24 | eleq2d 2673 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ (Vtx‘𝐺) ↔ 𝐾 ∈ 𝑉)) |
26 | 23, 25 | mpbird 246 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (Vtx‘𝐺)) |
27 | eqid 2610 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
28 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
29 | eqid 2610 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
30 | 27, 28, 29 | vtxd0nedgb 40703 | . . 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 246 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 {csn 4125 〈cop 4131 dom cdm 5038 ‘cfv 5804 0cc0 9815 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-vtxdg 40682 |
This theorem is referenced by: 1egrvtxdg0 40727 eupth2lem3lem3 41398 |
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