![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29005. (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 5425 | . . . . . . . 8 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4659 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) |
5 | 1, 4 | fsnd 6869 | . . . . 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 4820 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | upgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | sseqtrdi 4027 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4601 | . . . . . . 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 28876 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | 5, 13 | fssd 6728 | . . . 4 ⊢ (𝜑 → {⟨𝐴, {𝐵, 𝐶}⟩}:{𝐴}⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
15 | 14 | ffdmd 6741 | . . 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 5898 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {⟨𝐴, {𝐵, 𝐶}⟩}) |
18 | 16, 17 | feq12d 6698 | . . 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 257 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
20 | 9 | 1vgrex 28766 | . . 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 28848 | . . 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 257 | 1 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 {csn 4623 {cpr 4625 ⟨cop 4629 class class class wbr 5141 dom cdm 5669 ⟶wf 6532 ‘cfv 6536 ≤ cle 11250 2c2 12268 ♯chash 14293 Vtxcvtx 28760 iEdgciedg 28761 UPGraphcupgr 28844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-hash 14294 df-upgr 28846 |
This theorem is referenced by: upgr1eop 28879 upgr1eopALT 28881 |
Copyright terms: Public domain | W3C validator |