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Theorem usgraeq12d 25891
 Description: Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
Assertion
Ref Expression
usgraeq12d (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 USGrph 𝐸𝑊 USGrph 𝐹))

Proof of Theorem usgraeq12d
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprr 792 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → 𝐸 = 𝐹)
2 dmeq 5246 . . . . 5 (𝐸 = 𝐹 → dom 𝐸 = dom 𝐹)
32adantl 481 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
43adantl 481 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → dom 𝐸 = dom 𝐹)
5 pweq 4111 . . . . . . 7 (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
65adantr 480 . . . . . 6 ((𝑉 = 𝑊𝐸 = 𝐹) → 𝒫 𝑉 = 𝒫 𝑊)
76adantl 481 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → 𝒫 𝑉 = 𝒫 𝑊)
87difeq1d 3689 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
98rabeqdv 3167 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2})
101, 4, 9f1eq123d 6044 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐹:dom 𝐹1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
11 isusgra 25873 . . 3 ((𝑉𝑋𝐸𝑌) → (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
1211adantr 480 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
13 eleq1 2676 . . . . . . . 8 (𝑉 = 𝑊 → (𝑉𝑋𝑊𝑋))
1413biimpd 218 . . . . . . 7 (𝑉 = 𝑊 → (𝑉𝑋𝑊𝑋))
1514adantr 480 . . . . . 6 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝑉𝑋𝑊𝑋))
1615com12 32 . . . . 5 (𝑉𝑋 → ((𝑉 = 𝑊𝐸 = 𝐹) → 𝑊𝑋))
1716adantr 480 . . . 4 ((𝑉𝑋𝐸𝑌) → ((𝑉 = 𝑊𝐸 = 𝐹) → 𝑊𝑋))
1817imp 444 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → 𝑊𝑋)
19 eleq1 2676 . . . . . . . 8 (𝐸 = 𝐹 → (𝐸𝑌𝐹𝑌))
2019biimpd 218 . . . . . . 7 (𝐸 = 𝐹 → (𝐸𝑌𝐹𝑌))
2120adantl 481 . . . . . 6 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝐸𝑌𝐹𝑌))
2221com12 32 . . . . 5 (𝐸𝑌 → ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐹𝑌))
2322adantl 481 . . . 4 ((𝑉𝑋𝐸𝑌) → ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐹𝑌))
2423imp 444 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → 𝐹𝑌)
25 isusgra 25873 . . 3 ((𝑊𝑋𝐹𝑌) → (𝑊 USGrph 𝐹𝐹:dom 𝐹1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
2618, 24, 25syl2anc 691 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑊 USGrph 𝐹𝐹:dom 𝐹1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
2710, 12, 263bitr4d 299 1 (((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 USGrph 𝐸𝑊 USGrph 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-usgra 25862 This theorem is referenced by: (None)
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