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 𝐸 → 𝑉 USGrph (𝐸 ↾ 𝐴))

Proof of Theorem usgrares

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgraf 25875	. . . 4 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
2		resss 5342	. . . . 5 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
3		dmss 5245	. . . . 5 ⊢ ((𝐸 ↾ 𝐴) ⊆ 𝐸 → dom (𝐸 ↾ 𝐴) ⊆ dom 𝐸)
4	2, 3	mp1i 13	. . . 4 ⊢ (𝑉 USGrph 𝐸 → dom (𝐸 ↾ 𝐴) ⊆ dom 𝐸)
5		f1ssres 6021	. . . 4 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ∧ dom (𝐸 ↾ 𝐴) ⊆ dom 𝐸) → (𝐸 ↾ dom (𝐸 ↾ 𝐴)):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
6	1, 4, 5	syl2anc 691	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝐸 ↾ dom (𝐸 ↾ 𝐴)):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
7		resdmres 5543	. . . 4 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴)
8		f1eq1 6009	. . . 4 ⊢ ((𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴) → ((𝐸 ↾ dom (𝐸 ↾ 𝐴)):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ (𝐸 ↾ 𝐴):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
9	7, 8	ax-mp 5	. . 3 ⊢ ((𝐸 ↾ dom (𝐸 ↾ 𝐴)):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ (𝐸 ↾ 𝐴):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
10	6, 9	sylib 207	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝐸 ↾ 𝐴):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
11		usgrav 25867	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
12		resexg 5362	. . . 4 ⊢ (𝐸 ∈ V → (𝐸 ↾ 𝐴) ∈ V)
13	12	anim2i 591	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V))
14		isusgra 25873	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V) → (𝑉 USGrph (𝐸 ↾ 𝐴) ↔ (𝐸 ↾ 𝐴):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
15	11, 13, 14	3syl 18	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝑉 USGrph (𝐸 ↾ 𝐴) ↔ (𝐸 ↾ 𝐴):dom (𝐸 ↾ 𝐴)–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
16	10, 15	mpbird 246	1 ⊢ (𝑉 USGrph 𝐸 → 𝑉 USGrph (𝐸 ↾ 𝐴))

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)