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Theorem isuspgr 40384
Description: The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Hypotheses
Ref Expression
isuspgr.v 𝑉 = (Vtx‘𝐺)
isuspgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isuspgr (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hints:   𝑈(𝑥)   𝐸(𝑥)

Proof of Theorem isuspgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uspgr 40382 . . 3 USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
21eleq2i 2679 . 2 (𝐺 ∈ USPGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}})
3 fveq2 6088 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuspgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2661 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5235 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2618 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5234 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2659 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6088 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuspgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2661 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4112 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 3688 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3166 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
165, 9, 15f1eq123d 6029 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
17 fvex 6098 . . . . . 6 (Vtx‘𝑔) ∈ V
1817a1i 11 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
19 fveq2 6088 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
20 fvex 6098 . . . . . . 7 (iEdg‘𝑔) ∈ V
2120a1i 11 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
22 fveq2 6088 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2322adantr 479 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
24 simpr 475 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2524dmeqd 5235 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
26 pweq 4110 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2726ad2antlr 758 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2827difeq1d 3688 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2928rabeqdv 3166 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
3024, 25, 29f1eq123d 6029 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
3121, 23, 30sbcied2 3439 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
3218, 19, 31sbcied2 3439 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
3332cbvabv 2733 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} = { ∣ (iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
3416, 33elab2g 3321 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
352, 34syl5bb 270 1 (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899  Vcvv 3172  [wsbc 3401  cdif 3536  c0 3873  𝒫 cpw 4107  {csn 4124   class class class wbr 4577  dom cdm 5028  1-1wf1 5787  cfv 5790  cle 9931  2c2 10917  #chash 12934  Vtxcvtx 40231  iEdgciedg 40232   USPGraph cuspgr 40380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fv 5798  df-uspgr 40382
This theorem is referenced by:  uspgrf  40386  uspgrushgr  40407  uspgrupgr  40408  uspgrupgrushgr  40409  usgruspgr  40410  uspgr1e  40472
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