Theorem isfusgr 27683

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 6771	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		isfusgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eqtr4di 2798	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4	3	eleq1d 2825	. 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin))
5		df-fusgr 27682	. 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
6	4, 5	elrab2 3629	1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ‘cfv 6432  Fincfn 8716  Vtxcvtx 27364  USGraphcusgr 27517  FinUSGraphcfusgr 27681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440  df-fusgr 27682

This theorem is referenced by:  fusgrvtxfi  27684  isfusgrf1  27685  isfusgrcl  27686  fusgrusgr  27687  opfusgr  27688  fusgredgfi  27690  fusgrfis  27695  cusgrsizeindslem  27816  cusgrsizeinds  27817  sizusglecusglem2  27827  fusgrmaxsize  27829  finrusgrfusgr  27930  rusgrnumwwlks  28335  rusgrnumwwlk  28336  frrusgrord0lem  28699  frrusgrord0  28700  clwlknon2num  28728  numclwlk1lem1  28729  numclwlk1lem2  28730  friendshipgt3  28758