Users' Mathboxes Mathbox for BTernaryTau < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lfuhgr Structured version   Visualization version   GIF version

Theorem lfuhgr 34108
Description: A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
Hypotheses
Ref Expression
lfuhgr.1 𝑉 = (Vtxβ€˜πΊ)
lfuhgr.2 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
lfuhgr (𝐺 ∈ UHGraph β†’ (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
Distinct variable groups:   π‘₯,𝐺   π‘₯,𝑉
Allowed substitution hint:   𝐼(π‘₯)

Proof of Theorem lfuhgr
StepHypRef Expression
1 edgval 28309 . . . . 5 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
2 lfuhgr.2 . . . . . 6 𝐼 = (iEdgβ€˜πΊ)
32rneqi 5937 . . . . 5 ran 𝐼 = ran (iEdgβ€˜πΊ)
41, 3eqtr4i 2764 . . . 4 (Edgβ€˜πΊ) = ran 𝐼
54sseq1i 4011 . . 3 ((Edgβ€˜πΊ) βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)})
62uhgrfun 28326 . . . . 5 (𝐺 ∈ UHGraph β†’ Fun 𝐼)
7 fdmrn 6750 . . . . . 6 (Fun 𝐼 ↔ 𝐼:dom 𝐼⟢ran 𝐼)
8 fss 6735 . . . . . . 7 ((𝐼:dom 𝐼⟢ran 𝐼 ∧ ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}) β†’ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)})
98ex 414 . . . . . 6 (𝐼:dom 𝐼⟢ran 𝐼 β†’ (ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} β†’ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}))
107, 9sylbi 216 . . . . 5 (Fun 𝐼 β†’ (ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} β†’ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}))
116, 10syl 17 . . . 4 (𝐺 ∈ UHGraph β†’ (ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} β†’ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}))
12 frn 6725 . . . 4 (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} β†’ ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)})
1311, 12impbid1 224 . . 3 (𝐺 ∈ UHGraph β†’ (ran 𝐼 βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}))
145, 13bitrid 283 . 2 (𝐺 ∈ UHGraph β†’ ((Edgβ€˜πΊ) βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)}))
15 uhgredgss 28391 . . . . 5 (𝐺 ∈ UHGraph β†’ (Edgβ€˜πΊ) βŠ† (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}))
1615difss2d 4135 . . . 4 (𝐺 ∈ UHGraph β†’ (Edgβ€˜πΊ) βŠ† 𝒫 (Vtxβ€˜πΊ))
17 lfuhgr.1 . . . . 5 𝑉 = (Vtxβ€˜πΊ)
1817pweqi 4619 . . . 4 𝒫 𝑉 = 𝒫 (Vtxβ€˜πΊ)
1916, 18sseqtrrdi 4034 . . 3 (𝐺 ∈ UHGraph β†’ (Edgβ€˜πΊ) βŠ† 𝒫 𝑉)
20 ssrab 4071 . . . 4 ((Edgβ€˜πΊ) βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ ((Edgβ€˜πΊ) βŠ† 𝒫 𝑉 ∧ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
2120baib 537 . . 3 ((Edgβ€˜πΊ) βŠ† 𝒫 𝑉 β†’ ((Edgβ€˜πΊ) βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
2219, 21syl 17 . 2 (𝐺 ∈ UHGraph β†’ ((Edgβ€˜πΊ) βŠ† {π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
2314, 22bitr3d 281 1 (𝐺 ∈ UHGraph β†’ (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629   class class class wbr 5149  dom cdm 5677  ran crn 5678  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544   ≀ cle 11249  2c2 12267  β™―chash 14290  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307  UHGraphcuhgr 28316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-edg 28308  df-uhgr 28318
This theorem is referenced by:  lfuhgr2  34109
  Copyright terms: Public domain W3C validator