Theorem isfusgr 27666

Description: The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
isfusgr	⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Proof of Theorem isfusgr

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6768	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		isfusgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eqtr4di 2797	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4	3	eleq1d 2824	. 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin))
5		df-fusgr 27665	. 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
6	4, 5	elrab2 3628	1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  Fincfn 8707  Vtxcvtx 27347  USGraphcusgr 27500  FinUSGraphcfusgr 27664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-fusgr 27665

This theorem is referenced by:  fusgrvtxfi  27667  isfusgrf1  27668  isfusgrcl  27669  fusgrusgr  27670  opfusgr  27671  fusgredgfi  27673  fusgrfis  27678  cusgrsizeindslem  27799  cusgrsizeinds  27800  sizusglecusglem2  27810  fusgrmaxsize  27812  finrusgrfusgr  27913  rusgrnumwwlks  28318  rusgrnumwwlk  28319  frrusgrord0lem  28682  frrusgrord0  28683  clwlknon2num  28711  numclwlk1lem1  28712  numclwlk1lem2  28713  friendshipgt3  28741