Theorem upgr1edc 16108

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 4324	. . . . . . . 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 3717	. . . . . . 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 5658	. . . . 5 |- ( ph -> { <. A , { B , C } >. } : { A } --> { { B , C } } )
9	2, 3	prssd 3852	. . . . . . . 8 |- ( ph -> { B , C } C_ V )
10		upgr1e.v	. . . . . . . 8 |- V = (Vtx ` G )
11	9, 10	sseqtrdi 3285	. . . . . . 7 |- ( ph -> { B , C } C_ (Vtx ` G ) )
12		elpwg 3676	. . . . . . . 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 16107	. . . . 5 |- ( ph -> { { B , C } } C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
17	8, 16	fssd 5521	. . . 4 |- ( ph -> { <. A , { B , C } >. } : { A } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
18	17	ffdmd 5533	. . 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 4957	. . . 4 |- ( ph -> dom (iEdg ` G ) = dom { <. A , { B , C } >. } )
21	19, 20	feq12d 5497	. . 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 16007	. . 3 |- ( B e. V -> G e. _V )
24		eqid 2232	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
25		eqid 2232	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
26	24, 25	isupgren 16082	. . 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 716  DECID wdc 842   = wceq 1398   e. wcel 2203   {crab 2524   _Vcvv 2812   C_ wss 3210   ~Pcpw 3668   {csn 3688   {cpr 3689   <.cop 3691   class class class wbr 4108   dom cdm 4748   -->wf 5347   ` cfv 5351   1oc1o 6639   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15999  iEdgciedg 16000  UPGraphcupgr 16078

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sub 8445  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-dec 9709  df-ndx 13207  df-slot 13208  df-base 13210  df-edgf 15992  df-vtx 16001  df-iedg 16002  df-upgren 16080

This theorem is referenced by:  upgr1eopdc  16110  upgr1een  16111