Theorem upgr1or2 15909

Description: An edge of an undirected pseudograph has one or two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
upgr1or2	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )

Proof of Theorem upgr1or2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 |- V = (Vtx ` G )
2		isupgr.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	upgrfnen 15906	. . . 4 |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
4	3	ffvelcdmda 5772	. . 3 |- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	4	3impa 1218	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6		breq1 4086	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 1o <-> ( E ` F ) ~~ 1o ) )
7		breq1 4086	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 2o <-> ( E ` F ) ~~ 2o ) )
8	6, 7	orbi12d 798	. . . 4 |- ( x = ( E ` F ) -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
9	8	elrab 2959	. . 3 |- ( ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( ( E ` F ) e. ~P V /\ ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
10	9	simprbi 275	. 2 |- ( ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )
11	5, 10	syl 14	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   ~Pcpw 3649   class class class wbr 4083   Fn wfn 5313   ` cfv 5318   1oc1o 6561   2oc2o 6562   ~~ cen 6893  Vtxcvtx 15821  iEdgciedg 15822  UPGraphcupgr 15899

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118

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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8327  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-dec 9587  df-ndx 13043  df-slot 13044  df-base 13046  df-edgf 15814  df-vtx 15823  df-iedg 15824  df-upgren 15901

This theorem is referenced by:  upgrfi  15910  upgrex  15911