Theorem grstructeld2dom 15891

Description: If any representation of a graph with vertices V and edges E is an element of an arbitrary class C, then any structure with base set V and value E in the slot for edge functions (which is such a representation of a graph with vertices V and edges E) is an element of this class C. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)

Hypotheses

Ref	Expression
gropeld.g	|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )
gropeld.v	|- ( ph -> V e. U )
gropeld.e	|- ( ph -> E e. W )
grstructeld.s	|- ( ph -> S e. X )
grstructeld.f	|- ( ph -> Fun ( S \ { (/) } ) )
grstructeld2dom.d	|- ( ph -> 2o ~<_ dom S )
grstructeld.b	|- ( ph -> ( Base ` S ) = V )
grstructeld.e	|- ( ph -> (.ef ` S ) = E )

Assertion

Ref	Expression
grstructeld2dom	|- ( ph -> S e. C )

Distinct variable groups:   C, g   g, E   g, V   ph, g   S, g

Allowed substitution hints:   U( g)   W( g)   X( g)

Proof of Theorem grstructeld2dom

Step	Hyp	Ref	Expression
1		gropeld.g	. . 3 |- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )
2		gropeld.v	. . 3 |- ( ph -> V e. U )
3		gropeld.e	. . 3 |- ( ph -> E e. W )
4		grstructeld.s	. . 3 |- ( ph -> S e. X )
5		grstructeld.f	. . 3 |- ( ph -> Fun ( S \ { (/) } ) )
6		grstructeld2dom.d	. . 3 |- ( ph -> 2o ~<_ dom S )
7		grstructeld.b	. . 3 |- ( ph -> ( Base ` S ) = V )
8		grstructeld.e	. . 3 |- ( ph -> (.ef ` S ) = E )
9	1, 2, 3, 4, 5, 6, 7, 8	grstructd2dom 15889	. 2 |- ( ph -> [. S / g ]. g e. C )
10		sbcel1v 3092	. 2 |- ( [. S / g ]. g e. C <-> S e. C )
11	9, 10	sylib 122	1 |- ( ph -> S e. C )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   e. wcel 2200   [.wsbc 3029   \ cdif 3195   (/)c0 3492   {csn 3667   class class class wbr 4086   dom cdm 4723   Fun wfun 5318   ` cfv 5324   2oc2o 6571   ~<_ cdom 6903   Basecbs 13072  .efcedgf 15845  Vtxcvtx 15853  iEdgciedg 15854

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

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-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-dom 6906  df-sub 8342  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-dec 9602  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856

This theorem is referenced by: (None)