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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 3964 | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∈ {𝑁}) |
| 3 | 1loopgruspgr.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 4 | snex 5436 | . . . . . 6 ⊢ {𝑁} ∈ V | |
| 5 | fvsng 7200 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝑁} ∈ V) → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) | |
| 6 | 3, 4, 5 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ({〈𝐴, {𝑁}〉}‘𝐴) = {𝑁}) |
| 7 | 6 | eleq2d 2827 | . . . 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 5916 | . . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝑁}〉}) |
| 11 | dmsnopg 6233 | . . . . . . 7 ⊢ ({𝑁} ∈ V → dom {〈𝐴, {𝑁}〉} = {𝐴}) | |
| 12 | 4, 11 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → dom {〈𝐴, {𝑁}〉} = {𝐴}) |
| 13 | 10, 12 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
| 14 | 9 | fveq1d 6908 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝑖)) |
| 15 | 14 | eleq2d 2827 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
| 16 | 13, 15 | rexeqbidv 3347 | . . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝐺)𝐾 ∈ ((iEdg‘𝐺)‘𝑖) ↔ ∃𝑖 ∈ {𝐴}𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖))) |
| 17 | fveq2 6906 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → ({〈𝐴, {𝑁}〉}‘𝑖) = ({〈𝐴, {𝑁}〉}‘𝐴)) | |
| 18 | 17 | eleq2d 2827 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝑖) ↔ 𝐾 ∈ ({〈𝐴, {𝑁}〉}‘𝐴))) |
| 19 | 18 | rexsng 4676 | . . . . 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 279 | . . 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 3963 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
| 24 | 1loopgruspgr.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
| 25 | 24 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ (Vtx‘𝐺) ↔ 𝐾 ∈ 𝑉)) |
| 26 | 23, 25 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (Vtx‘𝐺)) |
| 27 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 28 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 29 | eqid 2737 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
| 30 | 27, 28, 29 | vtxd0nedgb 29506 | . . 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 257 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 {csn 4626 〈cop 4632 dom cdm 5685 ‘cfv 6561 0cc0 11155 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-hash 14370 df-vtxdg 29484 |
| This theorem is referenced by: 1egrvtxdg0 29529 eupth2lem3lem3 30249 |
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