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Theorem isumgr 26872
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 26860 . . 3 UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
21eleq2i 2902 . 2 (𝐺 ∈ UMGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}})
3 fveq2 6663 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isumgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2872 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5767 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2828 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5766 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2870 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6663 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isumgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2872 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4542 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4096 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3483 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
165, 9, 15feq123d 6496 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
17 fvexd 6678 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6663 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvexd 6678 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
20 fveq2 6663 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2120adantr 483 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
22 simpr 487 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2322dmeqd 5767 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
24 pweq 4540 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524ad2antlr 725 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2625difeq1d 4096 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2726rabeqdv 3483 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2})
2822, 23, 27feq123d 6496 . . . . . 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 3813 . . . . 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 3813 . . . 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 2887 . . 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 3666 . 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, 32syl5bb 285 1 (𝐺𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  {cab 2797  {crab 3140  Vcvv 3493  [wsbc 3770  cdif 3931  c0 4289  𝒫 cpw 4537  {csn 4559  dom cdm 5548  wf 6344  cfv 6348  2c2 11684  chash 13682  Vtxcvtx 26773  iEdgciedg 26774  UMGraphcumgr 26858
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-umgr 26860
This theorem is referenced by:  isumgrs  26873  umgrupgr  26880  umgr0e  26887  umgrislfupgr  26900
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