Theorem upgr1edc 15906

Description: A pseudograph with one edge. Such a graph is actually a simple pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)

Hypotheses

Ref	Expression
upgr1e.v	|- V = (Vtx ` G )
upgr1e.a	|- ( ph -> A e. X )
upgr1e.b	|- ( ph -> B e. V )
upgr1e.c	|- ( ph -> C e. V )
upgr1edc.dc	|- ( ph -> DECID B = C )
upgr1e.e	|- ( ph -> (iEdg ` G ) = { <. A , { B , C } >. } )

Assertion

Ref	Expression
upgr1edc	|- ( ph -> G e. UPGraph )

Proof of Theorem upgr1edc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr1e.a	. . . . . 6 |- ( ph -> A e. X )
2		upgr1e.b	. . . . . . . 8 |- ( ph -> B e. V )
3		upgr1e.c	. . . . . . . 8 |- ( ph -> C e. V )
4		prexg 4294	. . . . . . . 8 |- ( ( B e. V /\ C e. V ) -> { B , C } e. _V )
5	2, 3, 4	syl2anc 411	. . . . . . 7 |- ( ph -> { B , C } e. _V )
6		snidg 3695	. . . . . . 7 |- ( { B , C } e. _V -> { B , C } e. { { B , C } } )
7	5, 6	syl 14	. . . . . 6 |- ( ph -> { B , C } e. { { B , C } } )
8	1, 7	fsnd 5612	. . . . 5 |- ( ph -> { <. A , { B , C } >. } : { A } --> { { B , C } } )
9	2, 3	prssd 3826	. . . . . . . 8 |- ( ph -> { B , C } C_ V )
10		upgr1e.v	. . . . . . . 8 |- V = (Vtx ` G )
11	9, 10	sseqtrdi 3272	. . . . . . 7 |- ( ph -> { B , C } C_ (Vtx ` G ) )
12		elpwg 3657	. . . . . . . 8 |- ( { B , C } e. _V -> ( { B , C } e. ~P (Vtx ` G ) <-> { B , C } C_ (Vtx ` G ) ) )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> ( { B , C } e. ~P (Vtx ` G ) <-> { B , C } C_ (Vtx ` G ) ) )
14	11, 13	mpbird 167	. . . . . 6 |- ( ph -> { B , C } e. ~P (Vtx ` G ) )
15		upgr1edc.dc	. . . . . 6 |- ( ph -> DECID B = C )
16	14, 2, 3, 15	upgr1elem1 15905	. . . . 5 |- ( ph -> { { B , C } } C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
17	8, 16	fssd 5482	. . . 4 |- ( ph -> { <. A , { B , C } >. } : { A } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
18	17	ffdmd 5491	. . 3 |- ( ph -> { <. A , { B , C } >. } : dom { <. A , { B , C } >. } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
19		upgr1e.e	. . . 4 |- ( ph -> (iEdg ` G ) = { <. A , { B , C } >. } )
20	19	dmeqd 4922	. . . 4 |- ( ph -> dom (iEdg ` G ) = dom { <. A , { B , C } >. } )
21	19, 20	feq12d 5459	. . 3 |- ( ph -> ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> { <. A , { B , C } >. } : dom { <. A , { B , C } >. } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
22	18, 21	mpbird 167	. 2 |- ( ph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
23	10	1vgrex 15806	. . 3 |- ( B e. V -> G e. _V )
24		eqid 2229	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
25		eqid 2229	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
26	24, 25	isupgren 15880	. . 3 |- ( G e. _V -> ( G e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
27	2, 23, 26	3syl 17	. 2 |- ( ph -> ( G e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
28	22, 27	mpbird 167	1 |- ( ph -> G e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   {csn 3666   {cpr 3667   <.cop 3669   class class class wbr 4082   dom cdm 4716   -->wf 5310   ` cfv 5314   1oc1o 6545   2oc2o 6546   ~~ cen 6875  Vtxcvtx 15798  iEdgciedg 15799  UPGraphcupgr 15876

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  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-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-1o 6552  df-2o 6553  df-er 6670  df-en 6878  df-sub 8307  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-dec 9567  df-ndx 13021  df-slot 13022  df-base 13024  df-edgf 15791  df-vtx 15800  df-iedg 15801  df-upgren 15878

This theorem is referenced by:  upgr1eopdc  15908