Theorem upgr1een 16106

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16103 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)

Hypotheses

Ref	Expression
upgr1een.k	|- ( ph -> K e. X )
upgr1een.v	|- ( ph -> V e. Y )
upgr1een.e	|- ( ph -> E e. ~P V )
upgr1een.2o	|- ( ph -> E ~~ 2o )

Assertion

Ref	Expression
upgr1een	|- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )

Proof of Theorem upgr1een

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr1een.2o	. . 3 |- ( ph -> E ~~ 2o )
2		en2 7064	. . 3 |- ( E ~~ 2o -> E. u E. v E = { u , v } )
3	1, 2	syl 14	. 2 |- ( ph -> E. u E. v E = { u , v } )
4		eqid 2232	. . . . 5 |- (Vtx ` <. V , { <. K , E >. } >. ) = (Vtx ` <. V , { <. K , E >. } >. )
5		upgr1een.k	. . . . . 6 |- ( ph -> K e. X )
6	5	adantr 276	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> K e. X )
7		upgr1een.e	. . . . . . . . 9 |- ( ph -> E e. ~P V )
8	7	elpwid 3679	. . . . . . . 8 |- ( ph -> E C_ V )
9	8	adantr 276	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> E C_ V )
10		vex 2815	. . . . . . . . 9 |- u e. _V
11	10	prid1 3796	. . . . . . . 8 |- u e. { u , v }
12		simpr 110	. . . . . . . 8 |- ( ( ph /\ E = { u , v } ) -> E = { u , v } )
13	11, 12	eleqtrrid 2322	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> u e. E )
14	9, 13	sseldd 3238	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> u e. V )
15		upgr1een.v	. . . . . . . 8 |- ( ph -> V e. Y )
16		opexg 4343	. . . . . . . . . 10 |- ( ( K e. X /\ E e. ~P V ) -> <. K , E >. e. _V )
17	5, 7, 16	syl2anc 411	. . . . . . . . 9 |- ( ph -> <. K , E >. e. _V )
18		snexg 4296	. . . . . . . . 9 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
19	17, 18	syl 14	. . . . . . . 8 |- ( ph -> { <. K , E >. } e. _V )
20		opvtxfv 16004	. . . . . . . 8 |- ( ( V e. Y /\ { <. K , E >. } e. _V ) -> (Vtx ` <. V , { <. K , E >. } >. ) = V )
21	15, 19, 20	syl2anc 411	. . . . . . 7 |- ( ph -> (Vtx ` <. V , { <. K , E >. } >. ) = V )
22	21	adantr 276	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> (Vtx ` <. V , { <. K , E >. } >. ) = V )
23	14, 22	eleqtrrd 2312	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> u e. (Vtx ` <. V , { <. K , E >. } >. ) )
24		vex 2815	. . . . . . . . 9 |- v e. _V
25	24	prid2 3797	. . . . . . . 8 |- v e. { u , v }
26	25, 12	eleqtrrid 2322	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> v e. E )
27	9, 26	sseldd 3238	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> v e. V )
28	27, 22	eleqtrrd 2312	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> v e. (Vtx ` <. V , { <. K , E >. } >. ) )
29	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ E = { u , v } ) -> E ~~ 2o )
30	12, 29	eqbrtrrd 4132	. . . . . . . 8 |- ( ( ph /\ E = { u , v } ) -> { u , v } ~~ 2o )
31		pr2ne 7488	. . . . . . . . 9 |- ( ( u e. _V /\ v e. _V ) -> ( { u , v } ~~ 2o <-> u =/= v ) )
32	31	el2v 2818	. . . . . . . 8 |- ( { u , v } ~~ 2o <-> u =/= v )
33	30, 32	sylib 122	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> u =/= v )
34	33	olcd 742	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> ( u = v \/ u =/= v ) )
35		dcne 2423	. . . . . 6 |- (DECID u = v <-> ( u = v \/ u =/= v ) )
36	34, 35	sylibr 134	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> DECID u = v )
37		opiedgfv 16007	. . . . . . . 8 |- ( ( V e. Y /\ { <. K , E >. } e. _V ) -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , E >. } )
38	15, 19, 37	syl2anc 411	. . . . . . 7 |- ( ph -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , E >. } )
39	38	adantr 276	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , E >. } )
40	12	opeq2d 3889	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> <. K , E >. = <. K , { u , v } >. )
41	40	sneqd 3701	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> { <. K , E >. } = { <. K , { u , v } >. } )
42	39, 41	eqtrd 2265	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , { u , v } >. } )
43	4, 6, 23, 28, 36, 42	upgr1edc 16103	. . . 4 |- ( ( ph /\ E = { u , v } ) -> <. V , { <. K , E >. } >. e. UPGraph )
44	43	ex 115	. . 3 |- ( ph -> ( E = { u , v } -> <. V , { <. K , E >. } >. e. UPGraph ) )
45	44	exlimdvv 1947	. 2 |- ( ph -> ( E. u E. v E = { u , v } -> <. V , { <. K , E >. } >. e. UPGraph ) )
46	3, 45	mpd 13	1 |- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2203   =/= wne 2412   _Vcvv 2812   C_ wss 3210   ~Pcpw 3668   {csn 3688   {cpr 3689   <.cop 3691   class class class wbr 4108   ` cfv 5351   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073

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 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-upgren 16075

This theorem is referenced by:  umgr1een  16107  p1evtxdeqfilem  16293  p1evtxdeqfi  16294