![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > umgrupgr | Structured version Visualization version GIF version |
Description: An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
umgrupgr | ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2724 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isumgr 28824 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
4 | id 22 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) | |
5 | 2re 12283 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
6 | 5 | leidi 11745 | . . . . . . . . . 10 ⊢ 2 ≤ 2 |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2) |
8 | breq1 5141 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → ((♯‘𝑥) ≤ 2 ↔ 2 ≤ 2)) | |
9 | 7, 8 | mpbird 257 | . . . . . . . 8 ⊢ ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2) |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)) |
11 | 10 | ss2rabi 4066 | . . . . . 6 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
12 | 11 | a1i 11 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
13 | 4, 12 | fssd 6725 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | 3, 13 | syl6bi 253 | . . 3 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
15 | 14 | pm2.43i 52 | . 2 ⊢ (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
16 | 1, 2 | isupgr 28813 | . 2 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
17 | 15, 16 | mpbird 257 | 1 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 ∖ cdif 3937 ⊆ wss 3940 ∅c0 4314 𝒫 cpw 4594 {csn 4620 class class class wbr 5138 dom cdm 5666 ⟶wf 6529 ‘cfv 6533 ≤ cle 11246 2c2 12264 ♯chash 14287 Vtxcvtx 28725 iEdgciedg 28726 UPGraphcupgr 28809 UMGraphcumgr 28810 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-i2m1 11174 ax-1ne0 11175 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-2 12272 df-upgr 28811 df-umgr 28812 |
This theorem is referenced by: umgruhgr 28833 upgr0e 28840 umgrislfupgr 28852 nbumgrvtx 29072 umgrwlknloop 29375 umgrwwlks2on 29680 umgr3v3e3cycl 29906 konigsberg 29979 |
Copyright terms: Public domain | W3C validator |