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Mirrors > Home > MPE Home > Th. List > usgredgreu | Structured version Visualization version GIF version |
Description: For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
Ref | Expression |
---|---|
usgredg3.v | ⊢ 𝑉 = (Vtx‘𝐺) |
usgredg3.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
usgredgreu | ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgredg3.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | usgredg3.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | usgredg4 27151 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}) |
4 | eqtr2 2759 | . . . . 5 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → {𝑌, 𝑦} = {𝑌, 𝑥}) | |
5 | vex 3401 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | vex 3401 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | preqr2 4732 | . . . . 5 ⊢ ({𝑌, 𝑦} = {𝑌, 𝑥} → 𝑦 = 𝑥) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥) |
9 | 8 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥)) |
10 | 9 | ralrimivva 3103 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∀𝑦 ∈ 𝑉 ∀𝑥 ∈ 𝑉 (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥)) |
11 | preq2 4622 | . . . 4 ⊢ (𝑦 = 𝑥 → {𝑌, 𝑦} = {𝑌, 𝑥}) | |
12 | 11 | eqeq2d 2749 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ (𝐸‘𝑋) = {𝑌, 𝑥})) |
13 | 12 | reu4 3628 | . 2 ⊢ (∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ∧ ∀𝑦 ∈ 𝑉 ∀𝑥 ∈ 𝑉 (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥))) |
14 | 3, 10, 13 | sylanbrc 586 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ∃!wreu 3055 {cpr 4515 dom cdm 5519 ‘cfv 6333 Vtxcvtx 26933 iEdgciedg 26934 USGraphcusgr 27086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-oadd 8128 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-dju 9396 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-hash 13776 df-edg 26985 df-umgr 27020 df-usgr 27088 |
This theorem is referenced by: usgredg2vtxeuALT 27156 usgredg2vlem1 27159 usgredg2vlem2 27160 |
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