Theorem edgupgren 16296

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 16214	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	eleq2d 2308	. . . 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 2238	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2238	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	upgrfen 16252	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7	6	frnd 5538	. . . . 5 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
8	7	sseld 3247	. . . 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 4128	. . . 4 |- ( x = E -> ( x ~~ 1o <-> E ~~ 1o ) )
12		breq1 4128	. . . 4 |- ( x = E -> ( x ~~ 2o <-> E ~~ 2o ) )
13	11, 12	orbi12d 805	. . 3 |- ( x = E -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( E ~~ 1o \/ E ~~ 2o ) ) )
14	13	elrab 2982	. 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 720   = wceq 1402   e. wcel 2209   {crab 2532   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   ran crn 4770   ` cfv 5372   1oc1o 6670   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UPGraphcupgr 16246

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-upgren 16248

This theorem is referenced by: (None)