Theorem edgupgren 15933

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 15854	. . . . 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 15891	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7	6	frnd 5482	. . . . 5 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
8	7	sseld 3223	. . . 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 4085	. . . 4 |- ( x = E -> ( x ~~ 1o <-> E ~~ 1o ) )
12		breq1 4085	. . . 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 2959	. 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 3649   class class class wbr 4082   dom cdm 4718   ran crn 4719   ` cfv 5317   1oc1o 6553   2oc2o 6554   ~~ cen 6883  Vtxcvtx 15807  iEdgciedg 15808  Edgcedg 15852  UPGraphcupgr 15885

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-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  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-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-fo 5323  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-sub 8315  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-dec 9575  df-ndx 13030  df-slot 13031  df-base 13033  df-edgf 15800  df-vtx 15809  df-iedg 15810  df-edg 15853  df-upgren 15887

This theorem is referenced by: (None)