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| Mirrors > Home > MPE Home > Th. List > 1hevtxdg0 | Structured version Visualization version GIF version | ||
| Description: The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) |
| Ref | Expression |
|---|---|
| 1hevtxdg0.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) |
| 1hevtxdg0.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
| 1hevtxdg0.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
| 1hevtxdg0.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 1hevtxdg0.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
| 1hevtxdg0.n | ⊢ (𝜑 → 𝐷 ∉ 𝐸) |
| Ref | Expression |
|---|---|
| 1hevtxdg0 | ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1hevtxdg0.n | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∉ 𝐸) | |
| 2 | df-nel 3047 | . . . . . . 7 ⊢ (𝐷 ∉ 𝐸 ↔ ¬ 𝐷 ∈ 𝐸) | |
| 3 | 1, 2 | sylib 218 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐸) |
| 4 | 1hevtxdg0.i | . . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) | |
| 5 | 4 | fveq1d 6908 | . . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐴) = ({〈𝐴, 𝐸〉}‘𝐴)) |
| 6 | 1hevtxdg0.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 7 | 1hevtxdg0.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
| 8 | fvsng 7200 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ({〈𝐴, 𝐸〉}‘𝐴) = 𝐸) | |
| 9 | 6, 7, 8 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → ({〈𝐴, 𝐸〉}‘𝐴) = 𝐸) |
| 10 | 5, 9 | eqtrd 2777 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐴) = 𝐸) |
| 11 | 3, 10 | neleqtrrd 2864 | . . . . 5 ⊢ (𝜑 → ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴)) |
| 12 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝐴)) | |
| 13 | 12 | eleq2d 2827 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
| 14 | 13 | notbid 318 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
| 15 | 14 | ralsng 4675 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
| 16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
| 17 | 11, 16 | mpbird 257 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
| 18 | 4 | dmeqd 5916 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, 𝐸〉}) |
| 19 | dmsnopg 6233 | . . . . . 6 ⊢ (𝐸 ∈ 𝑌 → dom {〈𝐴, 𝐸〉} = {𝐴}) | |
| 20 | 7, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → dom {〈𝐴, 𝐸〉} = {𝐴}) |
| 21 | 18, 20 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
| 22 | 17, 21 | raleqtrrdv 3330 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
| 23 | ralnex 3072 | . . 3 ⊢ (∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) | |
| 24 | 22, 23 | sylib 218 | . 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
| 25 | 1hevtxdg0.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
| 26 | 1hevtxdg0.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
| 27 | 26 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ (Vtx‘𝐺) ↔ 𝐷 ∈ 𝑉)) |
| 28 | 25, 27 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (Vtx‘𝐺)) |
| 29 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 30 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 31 | eqid 2737 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
| 32 | 29, 30, 31 | vtxd0nedgb 29506 | . . 3 ⊢ (𝐷 ∈ (Vtx‘𝐺) → (((VtxDeg‘𝐺)‘𝐷) = 0 ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥))) |
| 33 | 28, 32 | syl 17 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝐷) = 0 ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥))) |
| 34 | 24, 33 | mpbird 257 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 ∃wrex 3070 {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: p1evtxdeq 29531 eupth2lem3lem6 30252 |
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