Theorem opvtxval 25883

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

Assertion

Ref	Expression
opvtxval	|- ( G e. ( _V X. _V ) -> (Vtx ` G ) = ( 1st ` G ) )

Proof of Theorem opvtxval

Step	Hyp	Ref	Expression
1		vtxval 25878	. 2 |- (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) )
2		iftrue 4092	. 2 |- ( G e. ( _V X. _V ) -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) = ( 1st ` G ) )
3	1, 2	syl5eq 2668	1 |- ( G e. ( _V X. _V ) -> (Vtx ` G ) = ( 1st ` G ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   X. cxp 5112   ` cfv 5888   1stc1st 7166   Basecbs 15857  Vtxcvtx 25874

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-vtx 25876

This theorem is referenced by:  opvtxfv  25884