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Mirrors > Home > MPE Home > Th. List > upgr1e | Structured version Visualization version GIF version |
Description: A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 26327. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.) |
Ref | Expression |
---|---|
upgr1e.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgr1e.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
upgr1e.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
upgr1e.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
upgr1e.e | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) |
Ref | Expression |
---|---|
upgr1e | ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgr1e.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | prex 5050 | . . . . . . . 8 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4345 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) |
5 | 1, 4 | fsnd 6332 | . . . . 5 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{{𝐵, 𝐶}}) |
6 | upgr1e.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
7 | upgr1e.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | 6, 7 | prssd 4491 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | upgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | syl6sseq 3784 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4300 | . . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
12 | 10, 11 | sylibr 224 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺)) |
13 | 12, 6 | upgr1elem 26198 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | 5, 13 | fssd 6210 | . . . 4 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
15 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ∈ V) |
16 | 15, 6 | upgr1elem 26198 | . . . . . . 7 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
17 | 5, 16 | fssd 6210 | . . . . . 6 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
18 | fdm 6204 | . . . . . 6 ⊢ ({〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → dom {〈𝐴, {𝐵, 𝐶}〉} = {𝐴}) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → dom {〈𝐴, {𝐵, 𝐶}〉} = {𝐴}) |
20 | 19 | feq2d 6184 | . . . 4 ⊢ (𝜑 → ({〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
21 | 14, 20 | mpbird 247 | . . 3 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
22 | upgr1e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) | |
23 | 22 | dmeqd 5473 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝐵, 𝐶}〉}) |
24 | 22, 23 | feq12d 6186 | . . 3 ⊢ (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ {〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
25 | 21, 24 | mpbird 247 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
26 | 9 | 1vgrex 26073 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
27 | eqid 2752 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
28 | eqid 2752 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
29 | 27, 28 | isupgr 26170 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
30 | 6, 26, 29 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
31 | 25, 30 | mpbird 247 | 1 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1624 ∈ wcel 2131 {crab 3046 Vcvv 3332 ∖ cdif 3704 ⊆ wss 3707 ∅c0 4050 𝒫 cpw 4294 {csn 4313 {cpr 4315 〈cop 4319 class class class wbr 4796 dom cdm 5258 ⟶wf 6037 ‘cfv 6041 ≤ cle 10259 2c2 11254 ♯chash 13303 Vtxcvtx 26065 iEdgciedg 26066 UPGraphcupgr 26166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8947 df-cda 9174 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-n0 11477 df-xnn0 11548 df-z 11562 df-uz 11872 df-fz 12512 df-hash 13304 df-upgr 26168 |
This theorem is referenced by: upgr1eop 26201 upgr1eopALT 26203 |
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