![]() |
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 2732 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2732 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isumgr 28110 | . . . 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 12237 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
6 | 5 | leidi 11699 | . . . . . . . . . 10 ⊢ 2 ≤ 2 |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2) |
8 | breq1 5114 | . . . . . . . . 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 4040 | . . . . . 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 6692 | . . . 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 28099 | . 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 1542 ∈ wcel 2107 {crab 3406 ∖ cdif 3911 ⊆ wss 3914 ∅c0 4288 𝒫 cpw 4566 {csn 4592 class class class wbr 5111 dom cdm 5639 ⟶wf 6498 ‘cfv 6502 ≤ cle 11200 2c2 12218 ♯chash 14241 Vtxcvtx 28011 iEdgciedg 28012 UPGraphcupgr 28095 UMGraphcumgr 28096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-i2m1 11129 ax-1ne0 11130 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7366 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-2 12226 df-upgr 28097 df-umgr 28098 |
This theorem is referenced by: umgruhgr 28119 upgr0e 28126 umgrislfupgr 28138 nbumgrvtx 28358 umgrwlknloop 28661 umgrwwlks2on 28966 umgr3v3e3cycl 29192 konigsberg 29265 |
Copyright terms: Public domain | W3C validator |