Theorem isfusgr 29350

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 6907	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		isfusgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eqtr4di 2793	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4	3	eleq1d 2824	. 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin))
5		df-fusgr 29349	. 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
6	4, 5	elrab2 3698	1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  Fincfn 8984  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-fusgr 29349

This theorem is referenced by:  fusgrvtxfi  29351  isfusgrf1  29352  isfusgrcl  29353  fusgrusgr  29354  opfusgr  29355  fusgredgfi  29357  fusgrfis  29362  cusgrsizeindslem  29484  cusgrsizeinds  29485  sizusglecusglem2  29495  fusgrmaxsize  29497  finrusgrfusgr  29598  rusgrnumwwlks  30004  rusgrnumwwlk  30005  frrusgrord0lem  30368  frrusgrord0  30369  clwlknon2num  30397  numclwlk1lem1  30398  numclwlk1lem2  30399  friendshipgt3  30427