Theorem upgruhgr 26879

Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)

Assertion

Ref	Expression
upgruhgr	⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Proof of Theorem upgruhgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2819	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2819	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 26863	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		ssrab2 4054	. . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		fss 6520	. . 3 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	3, 4, 5	sylancl 588	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	1, 2	isuhgr 26837	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	6, 7	mpbird 259	1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ∈ wcel 2108  {crab 3140   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  {csn 4559   class class class wbr 5057  dom cdm 5548  ⟶wf 6344  ‘cfv 6348   ≤ cle 10668  2c2 11684  ♯chash 13682  Vtxcvtx 26773  iEdgciedg 26774  UHGraphcuhgr 26833  UPGraphcupgr 26857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26835  df-upgr 26859

This theorem is referenced by:  umgruhgr  26881  upgrle2  26882  edglnl  26920  numedglnl  26921  usgruhgr  26960  subupgr  27061  upgrspan  27067  upgrreslem  27078  upgrres  27080  finsumvtxdg2ssteplem1  27319  finsumvtxdg2size  27324  upgrewlkle2  27380  upgredginwlk  27409  wlkiswwlks1  27637  wlkiswwlksupgr2  27647  eulerpathpr  28011  eulercrct  28013  upgracycumgr  32393  isomuspgrlem2c  43985