Theorem upgr1or2 15958

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 15955	. . . 4 |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
4	3	ffvelcdmda 5782	. . 3 |- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	4	3impa 1220	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6		breq1 4091	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 1o <-> ( E ` F ) ~~ 1o ) )
7		breq1 4091	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 2o <-> ( E ` F ) ~~ 2o ) )
8	6, 7	orbi12d 800	. . . 4 |- ( x = ( E ` F ) -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
9	8	elrab 2962	. . 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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   ~Pcpw 3652   class class class wbr 4088   Fn wfn 5321   ` cfv 5326   1oc1o 6575   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15869  iEdgciedg 15870  UPGraphcupgr 15948

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872  df-upgren 15950

This theorem is referenced by:  upgrfi  15959  upgrex  15960  subupgr  16130