Theorem upgr1een 16136

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16133 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 7067	. . 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 2234	. . . . 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 3682	. . . . . . . 8 |- ( ph -> E C_ V )
9	8	adantr 276	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> E C_ V )
10		vex 2818	. . . . . . . . 9 |- u e. _V
11	10	prid1 3799	. . . . . . . 8 |- u e. { u , v }
12		simpr 110	. . . . . . . 8 |- ( ( ph /\ E = { u , v } ) -> E = { u , v } )
13	11, 12	eleqtrrid 2324	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> u e. E )
14	9, 13	sseldd 3241	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> u e. V )
15		upgr1een.v	. . . . . . . 8 |- ( ph -> V e. Y )
16		opexg 4346	. . . . . . . . . 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 4299	. . . . . . . . 9 |- ( <. K , E >. e. _V -> { <. K , E >. } e. _V )
19	17, 18	syl 14	. . . . . . . 8 |- ( ph -> { <. K , E >. } e. _V )
20		opvtxfv 16034	. . . . . . . 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 2314	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> u e. (Vtx ` <. V , { <. K , E >. } >. ) )
24		vex 2818	. . . . . . . . 9 |- v e. _V
25	24	prid2 3800	. . . . . . . 8 |- v e. { u , v }
26	25, 12	eleqtrrid 2324	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> v e. E )
27	9, 26	sseldd 3241	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> v e. V )
28	27, 22	eleqtrrd 2314	. . . . 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 4135	. . . . . . . 8 |- ( ( ph /\ E = { u , v } ) -> { u , v } ~~ 2o )
31		pr2ne 7491	. . . . . . . . 9 |- ( ( u e. _V /\ v e. _V ) -> ( { u , v } ~~ 2o <-> u =/= v ) )
32	31	el2v 2821	. . . . . . . 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 2425	. . . . . 6 |- (DECID u = v <-> ( u = v \/ u =/= v ) )
36	34, 35	sylibr 134	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> DECID u = v )
37		opiedgfv 16037	. . . . . . . 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 3892	. . . . . . 7 |- ( ( ph /\ E = { u , v } ) -> <. K , E >. = <. K , { u , v } >. )
41	40	sneqd 3704	. . . . . 6 |- ( ( ph /\ E = { u , v } ) -> { <. K , E >. } = { <. K , { u , v } >. } )
42	39, 41	eqtrd 2267	. . . . 5 |- ( ( ph /\ E = { u , v } ) -> (iEdg ` <. V , { <. K , E >. } >. ) = { <. K , { u , v } >. } )
43	4, 6, 23, 28, 36, 42	upgr1edc 16133	. . . 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 1949	. 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 2205   =/= wne 2414   _Vcvv 2815   C_ wss 3213   ~Pcpw 3671   {csn 3691   {cpr 3692   <.cop 3694   class class class wbr 4111   ` cfv 5354   2oc2o 6643   ~~ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-upgren 16105

This theorem is referenced by:  umgr1een  16137  p1evtxdeqfilem  16323  p1evtxdeqfi  16324