Theorem upgr1edc 16042

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 4307	. . . . . . . 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 3702	. . . . . . 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 5637	. . . . 5 |- ( ph -> { <. A , { B , C } >. } : { A } --> { { B , C } } )
9	2, 3	prssd 3837	. . . . . . . 8 |- ( ph -> { B , C } C_ V )
10		upgr1e.v	. . . . . . . 8 |- V = (Vtx ` G )
11	9, 10	sseqtrdi 3276	. . . . . . 7 |- ( ph -> { B , C } C_ (Vtx ` G ) )
12		elpwg 3664	. . . . . . . 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 16041	. . . . 5 |- ( ph -> { { B , C } } C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
17	8, 16	fssd 5502	. . . 4 |- ( ph -> { <. A , { B , C } >. } : { A } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
18	17	ffdmd 5514	. . 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 4939	. . . 4 |- ( ph -> dom (iEdg ` G ) = dom { <. A , { B , C } >. } )
21	19, 20	feq12d 5479	. . 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 15941	. . 3 |- ( B e. V -> G e. _V )
24		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
25		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
26	24, 25	isupgren 16016	. . 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 2202   {crab 2515   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   {csn 3673   {cpr 3674   <.cop 3676   class class class wbr 4093   dom cdm 4731   -->wf 5329   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15933  iEdgciedg 15934  UPGraphcupgr 16012

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sub 8395  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-dec 9655  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15926  df-vtx 15935  df-iedg 15936  df-upgren 16014

This theorem is referenced by:  upgr1eopdc  16044  upgr1een  16045