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Theorem isuspgr 26324
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 26322 . . 3 USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
21eleq2i 2835 . 2 (𝐺 ∈ USPGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}})
3 fveq2 6374 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuspgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2816 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5493 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2773 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5492 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2814 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6374 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuspgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2816 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4319 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 3888 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3342 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
165, 9, 15f1eq123d 6313 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
17 fvexd 6389 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6374 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvexd 6389 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
20 fveq2 6374 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2120adantr 472 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
22 simpr 477 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2322dmeqd 5493 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
24 pweq 4317 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524ad2antlr 718 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2625difeq1d 3888 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2726rabeqdv 3342 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2822, 23, 27f1eq123d 6313 . . . . . 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}))
2919, 21, 28sbcied2 3633 . . . . 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}))
3017, 18, 29sbcied2 3633 . . . 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}))
3130cbvabv 2889 . . 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}}
3216, 31elab2g 3507 . 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}))
332, 32syl5bb 274 1 (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2750  {crab 3058  Vcvv 3349  [wsbc 3595  cdif 3728  c0 4078  𝒫 cpw 4314  {csn 4333   class class class wbr 4808  dom cdm 5276  1-1wf1 6064  cfv 6067  cle 10328  2c2 11326  chash 13320  Vtxcvtx 26164  iEdgciedg 26165  USPGraphcuspgr 26320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-nul 4948
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fv 6075  df-uspgr 26322
This theorem is referenced by:  uspgrf  26326  isuspgrop  26333  uspgrushgr  26347  uspgrupgr  26348  uspgrupgrushgr  26349  usgruspgr  26350  uspgr1e  26414
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