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Theorem umgrupgr 29121
Description: An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
Assertion
Ref Expression
umgrupgr (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)

Proof of Theorem umgrupgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2736 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isumgr 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
65leidi 11798 . . . . . . . . . 10 2 ≤ 2
76a1i 11 . . . . . . . . 9 ((♯‘𝑥) = 2 → 2 ≤ 2)
8 breq1 5145 . . . . . . . . 9 ((♯‘𝑥) = 2 → ((♯‘𝑥) ≤ 2 ↔ 2 ≤ 2))
97, 8mpbird 257 . . . . . . . 8 ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)
109a1i 11 . . . . . . 7 (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2))
1110ss2rabi 4076 . . . . . 6 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
1211a1i 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})
134, 12fssd 6752 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
143, 13biimtrdi 253 . . 3 (𝐺 ∈ UMGraph → (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1514pm2.43i 52 . 2 (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
161, 2isupgr 29102 . 2 (𝐺 ∈ UMGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1715, 16mpbird 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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