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Theorem isumgr 28355
Description: The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
Hypotheses
Ref Expression
isumgr.v 𝑉 = (Vtx‘𝐺)
isumgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isumgr (𝐺𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hints:   𝑈(𝑥)   𝐸(𝑥)

Proof of Theorem isumgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-umgr 28343 . . 3 UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
21eleq2i 2826 . 2 (𝐺 ∈ UMGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}})
3 fveq2 6892 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isumgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4eqtr4di 2791 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5906 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2742 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5905 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8eqtrdi 2789 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6892 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isumgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11eqtr4di 2791 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4620 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4122 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3448 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
165, 9, 15feq123d 6707 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
17 fvexd 6907 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6892 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvexd 6907 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
20 fveq2 6892 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2120adantr 482 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
22 simpr 486 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2322dmeqd 5906 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
24 pweq 4617 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524ad2antlr 726 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2625difeq1d 4122 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2726rabeqdv 3448 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2})
2822, 23, 27feq123d 6707 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
2919, 21, 28sbcied2 3825 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
3017, 18, 29sbcied2 3825 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
3130cbvabv 2806 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} = { ∣ (iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2}}
3216, 31elab2g 3671 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
332, 32bitrid 283 1 (𝐺𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  {crab 3433  Vcvv 3475  [wsbc 3778  cdif 3946  c0 4323  𝒫 cpw 4603  {csn 4629  dom cdm 5677  wf 6540  cfv 6544  2c2 12267  chash 14290  Vtxcvtx 28256  iEdgciedg 28257  UMGraphcumgr 28341
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-ext 2704  ax-nul 5307
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  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-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-umgr 28343
This theorem is referenced by:  isumgrs  28356  umgrupgr  28363  umgr0e  28370  umgrislfupgr  28383
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