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Theorem usgrares 25898
 Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 25853. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
usgrares (𝑉 USGrph 𝐸𝑉 USGrph (𝐸𝐴))

Proof of Theorem usgrares
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 usgraf 25875 . . . 4 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
2 resss 5342 . . . . 5 (𝐸𝐴) ⊆ 𝐸
3 dmss 5245 . . . . 5 ((𝐸𝐴) ⊆ 𝐸 → dom (𝐸𝐴) ⊆ dom 𝐸)
42, 3mp1i 13 . . . 4 (𝑉 USGrph 𝐸 → dom (𝐸𝐴) ⊆ dom 𝐸)
5 f1ssres 6021 . . . 4 ((𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ∧ dom (𝐸𝐴) ⊆ dom 𝐸) → (𝐸 ↾ dom (𝐸𝐴)):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
61, 4, 5syl2anc 691 . . 3 (𝑉 USGrph 𝐸 → (𝐸 ↾ dom (𝐸𝐴)):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
7 resdmres 5543 . . . 4 (𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴)
8 f1eq1 6009 . . . 4 ((𝐸 ↾ dom (𝐸𝐴)) = (𝐸𝐴) → ((𝐸 ↾ dom (𝐸𝐴)):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ (𝐸𝐴):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
97, 8ax-mp 5 . . 3 ((𝐸 ↾ dom (𝐸𝐴)):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ (𝐸𝐴):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
106, 9sylib 207 . 2 (𝑉 USGrph 𝐸 → (𝐸𝐴):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
11 usgrav 25867 . . 3 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
12 resexg 5362 . . . 4 (𝐸 ∈ V → (𝐸𝐴) ∈ V)
1312anim2i 591 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ (𝐸𝐴) ∈ V))
14 isusgra 25873 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐴) ∈ V) → (𝑉 USGrph (𝐸𝐴) ↔ (𝐸𝐴):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
1511, 13, 143syl 18 . 2 (𝑉 USGrph 𝐸 → (𝑉 USGrph (𝐸𝐴) ↔ (𝐸𝐴):dom (𝐸𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
1610, 15mpbird 246 1 (𝑉 USGrph 𝐸𝑉 USGrph (𝐸𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  –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-ral 2901  df-rex 2902  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-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  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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