Theorem isfusgr 26210

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 e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )

Proof of Theorem isfusgr

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
2		isfusgr.v	. . . 4 |- V = (Vtx ` G )
3	1, 2	syl6eqr 2674	. . 3 |- ( g = G -> (Vtx ` g ) = V )
4	3	eleq1d 2686	. 2 |- ( g = G -> ( (Vtx ` g ) e. Fin <-> V e. Fin ) )
5		df-fusgr 26209	. 2 |- FinUSGraph = { g e. USGraph | (Vtx ` g ) e. Fin }
6	4, 5	elrab2 3366	1 |- ( G e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888   Fincfn 7955  Vtxcvtx 25874   USGraph cusgr 26044   FinUSGraph cfusgr 26208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-fusgr 26209

This theorem is referenced by:  fusgrvtxfi  26211  isfusgrf1  26212  isfusgrcl  26213  fusgrusgr  26214  opfusgr  26215  fusgredgfi  26217  fusgrfis  26222  nbfusgrlevtxm1  26279  nbfusgrlevtxm2  26280  cusgrsizeindslem  26347  cusgrsizeinds  26348  sizusglecusglem2  26358  fusgrmaxsize  26360  finrusgrfusgr  26461  rusgrnumwwlks  26869  rusgrnumwwlk  26870  frrusgrord0lem  27203  frrusgrord0  27204  friendshipgt3  27256