Theorem isfusgr 29476

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 6862	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		isfusgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eqtr4di 2814	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4	3	eleq1d 2846	. 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin))
5		df-fusgr 29475	. 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
6	4, 5	elrab2 3652	1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ‘cfv 6516  Fincfn 8921  Vtxcvtx 29154  USGraphcusgr 29307  FinUSGraphcfusgr 29474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-fusgr 29475

This theorem is referenced by:  fusgrvtxfi  29477  isfusgrf1  29478  isfusgrcl  29479  fusgrusgr  29480  opfusgr  29481  fusgredgfi  29483  fusgrfis  29488  cusgrsizeindslem  29609  cusgrsizeinds  29610  sizusglecusglem2  29620  fusgrmaxsize  29622  finrusgrfusgr  29723  rusgrnumwwlks  30134  rusgrnumwwlk  30135  frrusgrord0lem  30498  frrusgrord0  30499  clwlknon2num  30527  numclwlk1lem1  30528  numclwlk1lem2  30529  friendshipgt3  30557