Theorem upgr1edc 16345

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 4347	. . . . . . . 8 |- ( ( B e. V /\ C e. V ) -> { B , C } e. _V )
5	2, 3, 4	syl2anc 415	. . . . . . 7 |- ( ph -> { B , C } e. _V )
6		snidg 3737	. . . . . . 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 5682	. . . . 5 |- ( ph -> { <. A , { B , C } >. } : { A } --> { { B , C } } )
9	2, 3	prssd 3872	. . . . . . . 8 |- ( ph -> { B , C } C_ V )
10		upgr1e.v	. . . . . . . 8 |- V = (Vtx ` G )
11	9, 10	sseqtrdi 3296	. . . . . . 7 |- ( ph -> { B , C } C_ (Vtx ` G ) )
12		elpwg 3696	. . . . . . . 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 16344	. . . . 5 |- ( ph -> { { B , C } } C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
17	8, 16	fssd 5545	. . . 4 |- ( ph -> { <. A , { B , C } >. } : { A } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
18	17	ffdmd 5557	. . 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 4981	. . . 4 |- ( ph -> dom (iEdg ` G ) = dom { <. A , { B , C } >. } )
21	19, 20	feq12d 5521	. . 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 16244	. . 3 |- ( B e. V -> G e. _V )
24		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
25		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
26	24, 25	isupgren 16319	. . 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   {csn 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   dom cdm 4772   -->wf 5371   ` cfv 5375   1oc1o 6674   2oc2o 6675   ~~ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315

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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  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-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-upgren 16317

This theorem is referenced by:  upgr1eopdc  16347  upgr1een  16348