Theorem opvtxfv 25884

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

Assertion

Ref	Expression
opvtxfv	|- ( ( V e. X /\ E e. Y ) -> (Vtx ` <. V , E >. ) = V )

Proof of Theorem opvtxfv

Step	Hyp	Ref	Expression
1		opelvvg 5165	. . 3 |- ( ( V e. X /\ E e. Y ) -> <. V , E >. e. ( _V X. _V ) )
2		opvtxval 25883	. . 3 |- ( <. V , E >. e. ( _V X. _V ) -> (Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
3	1, 2	syl 17	. 2 |- ( ( V e. X /\ E e. Y ) -> (Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
4		op1stg 7180	. 2 |- ( ( V e. X /\ E e. Y ) -> ( 1st ` <. V , E >. ) = V )
5	3, 4	eqtrd 2656	1 |- ( ( V e. X /\ E e. Y ) -> (Vtx ` <. V , E >. ) = V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   ` cfv 5888   1stc1st 7166  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  ax-un 6949

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-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-vtx 25876

This theorem is referenced by:  opvtxov  25885  opvtxfvi  25889  graop  25921  gropd  25923  isuhgrop  25965  uhgrunop  25970  upgrop  25989  upgr1eop  26010  upgrunop  26014  umgrunop  26016  isuspgrop  26056  isusgrop  26057  ausgrusgrb  26060  uspgr1eop  26139  usgr1eop  26142  usgrexmpllem  26152  uhgrspanop  26188  uhgrspan1lem2  26193  upgrres1lem2  26203  opfusgr  26215  fusgrfisbase  26220  fusgrfisstep  26221  usgrexi  26337  cusgrexi  26339  fusgrmaxsize  26360  p1evtxdeqlem  26408  p1evtxdeq  26409  p1evtxdp1  26410  uspgrloopvtx  26411  umgr2v2evtx  26417  wlk2v2e  27017  eupthvdres  27095  eupth2lemb  27097  konigsbergvtx  27106  konigsberg  27119  uspgrsprfo  41756