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Theorem isushgr 25727
 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 25725 . . 3 USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
21eleq2i 2680 . 2 (𝐺 ∈ USHGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})})
3 fveq2 6103 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuhgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2662 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5248 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2619 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5247 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2660 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6103 . . . . . . 7 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuhgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2662 . . . . . 6 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4113 . . . . 5 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 3689 . . . 4 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
155, 9, 14f1eq123d 6044 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅}) ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
16 fvex 6113 . . . . . 6 (Vtx‘𝑔) ∈ V
1716a1i 11 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6103 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvex 6113 . . . . . . 7 (iEdg‘𝑔) ∈ V
2019a1i 11 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
21 fveq2 6103 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2221adantr 480 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
23 simpr 476 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2423dmeqd 5248 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
25 simpr 476 . . . . . . . . . 10 ((𝑔 = 𝑣 = (Vtx‘)) → 𝑣 = (Vtx‘))
2625pweqd 4113 . . . . . . . . 9 ((𝑔 = 𝑣 = (Vtx‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2726difeq1d 3689 . . . . . . . 8 ((𝑔 = 𝑣 = (Vtx‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2827adantr 480 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2923, 24, 28f1eq123d 6044 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
3020, 22, 29sbcied2 3440 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
3117, 18, 30sbcied2 3440 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})))
3231cbvabv 2734 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})} = { ∣ (iEdg‘):dom (iEdg‘)–1-1→(𝒫 (Vtx‘) ∖ {∅})}
3315, 32elab2g 3322 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
342, 33syl5bb 271 1 (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   USHGraph cushgr 25723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-ushgr 25725 This theorem is referenced by:  ushgrf  25729  uspgrushgr  40405
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