Theorem umgredgprv 16127

Description: In a multigraph, an edge is an unordered pair of vertices. This theorem would not hold for arbitrary hyper-/pseudographs since either M or N could be proper classes ( ( E ` X ) would be a loop in this case), which are no vertices of course. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.)

Hypotheses

Ref	Expression
umgrnloopv.e	|- E = (iEdg ` G )
umgredgprv.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
umgredgprv	|- ( ( G e. UMGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )

Proof of Theorem umgredgprv

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( E ` X ) = { M , N } )
2		umgruhgr 16125	. . . . . 6 |- ( G e. UMGraph -> G e. UHGraph )
3		umgredgprv.v	. . . . . . 7 |- V = (Vtx ` G )
4		umgrnloopv.e	. . . . . . 7 |- E = (iEdg ` G )
5	3, 4	uhgrss 16087	. . . . . 6 |- ( ( G e. UHGraph /\ X e. dom E ) -> ( E ` X ) C_ V )
6	2, 5	sylan 283	. . . . 5 |- ( ( G e. UMGraph /\ X e. dom E ) -> ( E ` X ) C_ V )
7	6	adantr 276	. . . 4 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( E ` X ) C_ V )
8	1, 7	eqsstrrd 3277	. . 3 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> { M , N } C_ V )
9	3, 4	umgredg2en 16121	. . . . . . 7 |- ( ( G e. UMGraph /\ X e. dom E ) -> ( E ` X ) ~~ 2o )
10	9	adantr 276	. . . . . 6 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( E ` X ) ~~ 2o )
11	1, 10	eqbrtrrd 4135	. . . . 5 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> { M , N } ~~ 2o )
12		pr2cv 7496	. . . . 5 |- ( { M , N } ~~ 2o -> ( M e. _V /\ N e. _V ) )
13	11, 12	syl 14	. . . 4 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( M e. _V /\ N e. _V ) )
14		prid1g 3797	. . . . 5 |- ( M e. _V -> M e. { M , N } )
15		prid2g 3798	. . . . 5 |- ( N e. _V -> N e. { M , N } )
16	14, 15	anim12i 338	. . . 4 |- ( ( M e. _V /\ N e. _V ) -> ( M e. { M , N } /\ N e. { M , N } ) )
17		prssg 3853	. . . 4 |- ( ( M e. { M , N } /\ N e. { M , N } ) -> ( ( M e. V /\ N e. V ) <-> { M , N } C_ V ) )
18	13, 16, 17	3syl 17	. . 3 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( ( M e. V /\ N e. V ) <-> { M , N } C_ V ) )
19	8, 18	mpbird 167	. 2 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> ( M e. V /\ N e. V ) )
20	19	ex 115	1 |- ( ( G e. UMGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   {cpr 3692   class class class wbr 4111   dom cdm 4751   ` cfv 5354   2oc2o 6643   ~~ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  UHGraphcuhgr 16079  UMGraphcumgr 16104

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-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-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-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-uhgrm 16081  df-upgren 16105  df-umgren 16106

This theorem is referenced by:  umgrnloop  16128  usgredgprv  16208