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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 29183. (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 5440 | . . . . . . . 8 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4669 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) |
5 | 1, 4 | fsnd 6888 | . . . . 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 4831 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | upgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | sseqtrdi 4030 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4611 | . . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
12 | 10, 11 | sylibr 233 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺)) |
13 | 12, 6 | upgr1elem 29051 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | 5, 13 | fssd 6747 | . . . 4 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
15 | 14 | ffdmd 6761 | . . 3 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
16 | upgr1e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) | |
17 | 16 | dmeqd 5914 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝐵, 𝐶}〉}) |
18 | 16, 17 | feq12d 6718 | . . 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})) |
19 | 15, 18 | mpbird 256 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
20 | 9 | 1vgrex 28941 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
21 | eqid 2726 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
22 | eqid 2726 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
23 | 21, 22 | isupgr 29023 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
24 | 6, 20, 23 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
25 | 19, 24 | mpbird 256 | 1 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 𝒫 cpw 4607 {csn 4633 {cpr 4635 〈cop 4639 class class class wbr 5155 dom cdm 5684 ⟶wf 6552 ‘cfv 6556 ≤ cle 11301 2c2 12321 ♯chash 14349 Vtxcvtx 28935 iEdgciedg 28936 UPGraphcupgr 29019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-oadd 8502 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-dju 9946 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12599 df-z 12613 df-uz 12877 df-fz 13541 df-hash 14350 df-upgr 29021 |
This theorem is referenced by: upgr1eop 29054 upgr1eopALT 29056 |
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