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Theorem isushgr 28186
Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
Hypotheses
Ref Expression
isuhgr.v 𝑉 = (Vtx‘𝐺)
isuhgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isushgr (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isushgr
Dummy variables 𝑔 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ushgr 28184 . . 3 USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
21eleq2i 2824 . 2 (𝐺 ∈ USHGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})})
3 fveq2 6878 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuhgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4eqtr4di 2789 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5897 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2740 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5896 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8eqtrdi 2787 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6878 . . . . . . 7 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuhgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11eqtr4di 2789 . . . . . 6 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4613 . . . . 5 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4117 . . . 4 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
155, 9, 14f1eq123d 6812 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅}) ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
16 fvexd 6893 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
17 fveq2 6878 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
18 fvexd 6893 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
19 fveq2 6878 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2019adantr 481 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
21 simpr 485 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2221dmeqd 5897 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
23 simpr 485 . . . . . . . . . 10 ((𝑔 = 𝑣 = (Vtx‘)) → 𝑣 = (Vtx‘))
2423pweqd 4613 . . . . . . . . 9 ((𝑔 = 𝑣 = (Vtx‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524difeq1d 4117 . . . . . . . 8 ((𝑔 = 𝑣 = (Vtx‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2625adantr 481 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2721, 22, 26f1eq123d 6812 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
2818, 20, 27sbcied2 3820 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
2916, 17, 28sbcied2 3820 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
3029cbvabv 2804 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})} = { ∣ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})}
3115, 30elab2g 3666 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
322, 31bitrid 282 1 (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2708  Vcvv 3473  [wsbc 3773  cdif 3941  c0 4318  𝒫 cpw 4596  {csn 4622  dom cdm 5669  1-1wf1 6529  cfv 6532  Vtxcvtx 28121  iEdgciedg 28122  USHGraphcushgr 28182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fv 6540  df-ushgr 28184
This theorem is referenced by:  ushgrf  28188  uspgrushgr  28300
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