Theorem edgupgren 15991

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 15909	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	eleq2d 2301	. . . 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 2231	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2231	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	upgrfen 15947	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7	6	frnd 5492	. . . . 5 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
8	7	sseld 3226	. . . 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 4091	. . . 4 |- ( x = E -> ( x ~~ 1o <-> E ~~ 1o ) )
12		breq1 4091	. . . 4 |- ( x = E -> ( x ~~ 2o <-> E ~~ 2o ) )
13	11, 12	orbi12d 800	. . 3 |- ( x = E -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( E ~~ 1o \/ E ~~ 2o ) ) )
14	13	elrab 2962	. 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 715   = wceq 1397   e. wcel 2202   {crab 2514   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   ran crn 4726   ` cfv 5326   1oc1o 6574   2oc2o 6575   ~~ cen 6906  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UPGraphcupgr 15941

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-upgren 15943

This theorem is referenced by: (None)