Theorem basvtxval2dom 15829

Description: The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)

Hypotheses

Ref	Expression
basvtxval.s	|- ( ph -> G Struct X )
basvtxval2dom.d	|- ( ph -> 2o ~<_ dom G )
basvtxval.v	|- ( ph -> V e. Y )
basvtxval.b	|- ( ph -> <. ( Base ` ndx ) , V >. e. G )

Assertion

Ref	Expression
basvtxval2dom	|- ( ph -> (Vtx ` G ) = V )

Proof of Theorem basvtxval2dom

Step	Hyp	Ref	Expression
1		basvtxval.s	. . . 4 |- ( ph -> G Struct X )
2		structex 13039	. . . 4 |- ( G Struct X -> G e. _V )
3	1, 2	syl 14	. . 3 |- ( ph -> G e. _V )
4		structn0fun 13040	. . . 4 |- ( G Struct X -> Fun ( G \ { (/) } ) )
5	1, 4	syl 14	. . 3 |- ( ph -> Fun ( G \ { (/) } ) )
6		basvtxval2dom.d	. . 3 |- ( ph -> 2o ~<_ dom G )
7		funvtxdm2domval 15824	. . 3 |- ( ( G e. _V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )
8	3, 5, 6, 7	syl3anc 1271	. 2 |- ( ph -> (Vtx ` G ) = ( Base ` G ) )
9		basvtxval.v	. . 3 |- ( ph -> V e. Y )
10		basvtxval.b	. . 3 |- ( ph -> <. ( Base ` ndx ) , V >. e. G )
11	1, 9, 10	opelstrbas 13143	. 2 |- ( ph -> V = ( Base ` G ) )
12	8, 11	eqtr4d 2265	1 |- ( ph -> (Vtx ` G ) = V )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   \ cdif 3194   (/)c0 3491   {csn 3666   <.cop 3669   class class class wbr 4082   dom cdm 4718   Fun wfun 5311   ` cfv 5317   2oc2o 6554   ~<_ cdom 6884   Struct cstr 13023   ndxcnx 13024   Basecbs 13027  Vtxcvtx 15807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-fv 5325  df-1st 6284  df-1o 6560  df-2o 6561  df-dom 6887  df-inn 9107  df-struct 13029  df-ndx 13030  df-slot 13031  df-base 13033  df-vtx 15809

This theorem is referenced by:  structvtxval  15834  structgrssvtx  15837