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Theorem isupgr 28848
Description: The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Hypotheses
Ref Expression
isupgr.v 𝑉 = (Vtx‘𝐺)
isupgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isupgr (𝐺𝑈 → (𝐺 ∈ UPGraph ↔ 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 isupgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-upgr 28846 . . 3 UPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
21eleq2i 2819 . 2 (𝐺 ∈ UPGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}})
3 fveq2 6884 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isupgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4eqtr4di 2784 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5898 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2735 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5897 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8eqtrdi 2782 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6884 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isupgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11eqtr4di 2784 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4614 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4116 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3441 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
165, 9, 15feq123d 6699 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
17 fvexd 6899 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6884 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvexd 6899 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
20 fveq2 6884 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2120adantr 480 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
22 simpr 484 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2322dmeqd 5898 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
24 pweq 4611 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524ad2antlr 724 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2625difeq1d 4116 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2726rabeqdv 3441 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2822, 23, 27feq123d 6699 . . . . . 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 3819 . . . . 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 3819 . . . 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 2799 . . 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 3665 . 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 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2703  {crab 3426  Vcvv 3468  [wsbc 3772  cdif 3940  c0 4317  𝒫 cpw 4597  {csn 4623   class class class wbr 5141  dom cdm 5669  wf 6532  cfv 6536  cle 11250  2c2 12268  chash 14293  Vtxcvtx 28760  iEdgciedg 28761  UPGraphcupgr 28844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-upgr 28846
This theorem is referenced by:  wrdupgr  28849  upgrf  28850  upgrop  28858  umgrupgr  28867  upgr1e  28877  upgrun  28882  uspgrupgr  28940  subupgr  29048  upgrres  29067  upgrres1  29074
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