Theorem upgr1edc 15764

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 4260	. . . . . . . 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 3664	. . . . . . 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 5575	. . . . 5 |- ( ph -> { <. A , { B , C } >. } : { A } --> { { B , C } } )
9	2, 3	prssd 3795	. . . . . . . 8 |- ( ph -> { B , C } C_ V )
10		upgr1e.v	. . . . . . . 8 |- V = (Vtx ` G )
11	9, 10	sseqtrdi 3243	. . . . . . 7 |- ( ph -> { B , C } C_ (Vtx ` G ) )
12		elpwg 3626	. . . . . . . 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 15763	. . . . 5 |- ( ph -> { { B , C } } C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
17	8, 16	fssd 5445	. . . 4 |- ( ph -> { <. A , { B , C } >. } : { A } --> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
18	17	ffdmd 5454	. . 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 4886	. . . 4 |- ( ph -> dom (iEdg ` G ) = dom { <. A , { B , C } >. } )
21	19, 20	feq12d 5422	. . 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 15669	. . 3 |- ( B e. V -> G e. _V )
24		eqid 2206	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
25		eqid 2206	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
26	24, 25	isupgren 15741	. . 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 710  DECID wdc 836   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773   C_ wss 3168   ~Pcpw 3618   {csn 3635   {cpr 3636   <.cop 3638   class class class wbr 4048   dom cdm 4680   -->wf 5273   ` cfv 5277   1oc1o 6505   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UPGraphcupgr 15737

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-upgren 15739

This theorem is referenced by:  upgr1eopdc  15766