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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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