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| 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 2736 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 3 | 1, 2 | isumgr 29113 | . . . 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 12341 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 6 | 5 | leidi 11798 | . . . . . . . . . 10 ⊢ 2 ≤ 2 | 
| 7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2) | 
| 8 | breq1 5145 | . . . . . . . . 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 4076 | . . . . . 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 6752 | . . . 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 | biimtrdi 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 29102 | . 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 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 {csn 4625 class class class wbr 5142 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 ≤ cle 11297 2c2 12322 ♯chash 14370 Vtxcvtx 29014 iEdgciedg 29015 UPGraphcupgr 29098 UMGraphcumgr 29099 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-i2m1 11224 ax-1ne0 11225 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 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-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-2 12330 df-upgr 29100 df-umgr 29101 | 
| This theorem is referenced by: umgruhgr 29122 upgr0e 29129 umgrislfupgr 29141 nbumgrvtx 29364 umgrwlknloop 29668 umgrwwlks2on 29978 umgr3v3e3cycl 30204 konigsberg 30277 | 
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