Theorem edgupgren 16136

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 16054	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	eleq2d 2302	. . . 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 2232	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2232	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	upgrfen 16092	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7	6	frnd 5518	. . . . 5 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
8	7	sseld 3237	. . . 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 4112	. . . 4 |- ( x = E -> ( x ~~ 1o <-> E ~~ 1o ) )
12		breq1 4112	. . . 4 |- ( x = E -> ( x ~~ 2o <-> E ~~ 2o ) )
13	11, 12	orbi12d 801	. . 3 |- ( x = E -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( E ~~ 1o \/ E ~~ 2o ) ) )
14	13	elrab 2973	. 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 716   = wceq 1398   e. wcel 2203   {crab 2524   ~Pcpw 3669   class class class wbr 4109   dom cdm 4749   ran crn 4750   ` cfv 5352   1oc1o 6640   2oc2o 6641   ~~ cen 6973  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  UPGraphcupgr 16086

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-upgren 16088

This theorem is referenced by: (None)