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 26136. (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 4909 | . . . . . . . 8 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4208 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) |
5 | 1, 4 | fsnd 6179 | . . . . 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 4354 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | upgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | syl6sseq 3651 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4164 | . . . . . . 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 26007 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
14 | 5, 13 | fssd 6057 | . . . 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 26007 | . . . . . . 7 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (V ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
17 | 5, 16 | fssd 6057 | . . . . . 6 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}⟶{𝑥 ∈ (V ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
18 | fdm 6051 | . . . . . 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 6031 | . . . 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 5326 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝐵, 𝐶}〉}) |
24 | 22, 23 | feq12d 6033 | . . 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 25882 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
27 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
28 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
29 | 27, 28 | isupgr 25979 | . . 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 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 〈cop 4183 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 ≤ cle 10075 2c2 11070 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 UPGraph cupgr 25975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-upgr 25977 |
This theorem is referenced by: upgr1eop 26010 upgr1eopALT 26012 |
Copyright terms: Public domain | W3C validator |