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Theorem isusgr 26269
Description: The property of being a simple graph. (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
isusgr (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 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 isusgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-usgr 26267 . . 3 USGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
21eleq2i 2841 . 2 (𝐺 ∈ USGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}})
3 fveq2 6332 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuspgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2822 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5464 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2779 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5463 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2820 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6332 . . . . . . . 8 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuspgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2822 . . . . . . 7 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4300 . . . . . 6 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 3876 . . . . 5 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1514rabeqdv 3343 . . . 4 ( = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
165, 9, 15f1eq123d 6272 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)–1-1→{𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
17 fvexd 6344 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
18 fveq2 6332 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fvexd 6344 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
20 fveq2 6332 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2120adantr 466 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
22 simpr 471 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2322dmeqd 5464 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
24 pweq 4298 . . . . . . . . . 10 (𝑣 = (Vtx‘) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524ad2antlr 698 . . . . . . . . 9 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2625difeq1d 3876 . . . . . . . 8 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2726rabeqdv 3343 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 (Vtx‘) ∖ {∅}) ∣ (♯‘𝑥) = 2})
2822, 23, 27f1eq123d 6272 . . . . . 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 3623 . . . . 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 3623 . . . 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 2895 . . 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 3502 . 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 272 1 (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  {cab 2756  {crab 3064  Vcvv 3349  [wsbc 3585  cdif 3718  c0 4061  𝒫 cpw 4295  {csn 4314  dom cdm 5249  1-1wf1 6028  cfv 6031  2c2 11271  chash 13320  Vtxcvtx 26094  iEdgciedg 26095  USGraphcusgr 26265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fv 6039  df-usgr 26267
This theorem is referenced by:  usgrf  26271  isusgrs  26272  usgruspgr  26294  usgrumgruspgr  26296  usgrislfuspgr  26300  usgr0e  26350  usgr0  26357
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