Theorem edgupgren 15980

Description: Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)

Assertion

Ref	Expression
edgupgren	|- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ ( E ~~ 1o \/ E ~~ 2o ) ) )

Proof of Theorem edgupgren

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		edgvalg 15900	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	eleq2d 2299	. . . 4 |- ( G e. UPGraph -> ( E e. (Edg ` G ) <-> E e. ran (iEdg ` G ) ) )
3	2	biimpa 296	. . 3 |- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> E e. ran (iEdg ` G ) )
4		eqid 2229	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2229	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	upgrfen 15938	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7	6	frnd 5489	. . . . 5 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
8	7	sseld 3224	. . . 4 |- ( G e. UPGraph -> ( E e. ran (iEdg ` G ) -> E e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
9	8	adantr 276	. . 3 |- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> ( E e. ran (iEdg ` G ) -> E e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
10	3, 9	mpd 13	. 2 |- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> E e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11		breq1 4089	. . . 4 |- ( x = E -> ( x ~~ 1o <-> E ~~ 1o ) )
12		breq1 4089	. . . 4 |- ( x = E -> ( x ~~ 2o <-> E ~~ 2o ) )
13	11, 12	orbi12d 798	. . 3 |- ( x = E -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( E ~~ 1o \/ E ~~ 2o ) ) )
14	13	elrab 2960	. 2 |- ( E e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( E e. ~P (Vtx ` G ) /\ ( E ~~ 1o \/ E ~~ 2o ) ) )
15	10, 14	sylib 122	1 |- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ ( E ~~ 1o \/ E ~~ 2o ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {crab 2512   ~Pcpw 3650   class class class wbr 4086   dom cdm 4723   ran crn 4724   ` cfv 5324   1oc1o 6570   2oc2o 6571   ~~ cen 6902  Vtxcvtx 15853  iEdgciedg 15854  Edgcedg 15898  UPGraphcupgr 15932

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-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-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-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-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  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  df-edg 15899  df-upgren 15934

This theorem is referenced by: (None)