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Theorem umgrupgr 26219
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 2771 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2771 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isumgr 26211 . . . 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 11296 . . . . . . . . . . 11 2 ∈ ℝ
65leidi 10768 . . . . . . . . . 10 2 ≤ 2
76a1i 11 . . . . . . . . 9 ((♯‘𝑥) = 2 → 2 ≤ 2)
8 breq1 4790 . . . . . . . . 9 ((♯‘𝑥) = 2 → ((♯‘𝑥) ≤ 2 ↔ 2 ≤ 2))
97, 8mpbird 247 . . . . . . . 8 ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)
109a1i 11 . . . . . . 7 (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2))
1110ss2rabi 3833 . . . . . 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 6198 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
143, 13syl6bi 243 . . 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 26200 . 2 (𝐺 ∈ UMGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1715, 16mpbird 247 1 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317   class class class wbr 4787  dom cdm 5250  wf 6026  cfv 6030  cle 10281  2c2 11276  chash 13321  Vtxcvtx 26095  iEdgciedg 26096  UPGraphcupgr 26196  UMGraphcumgr 26197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-i2m1 10210  ax-1ne0 10211  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-2 11285  df-upgr 26198  df-umgr 26199
This theorem is referenced by:  umgruhgr  26220  upgr0e  26227  umgrislfupgr  26239  nbumgrvtx  26465  umgrwlknloop  26780  umgrwwlks2on  27105  umgr3v3e3cycl  27364  konigsberg  27437
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