Theorem umgredgprv 16360

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 16358	. . . . . 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 16320	. . . . . 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 3285	. . 3 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> { M , N } C_ V )
9	3, 4	umgredg2en 16354	. . . . . . 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 4154	. . . . 5 |- ( ( ( G e. UMGraph /\ X e. dom E ) /\ ( E ` X ) = { M , N } ) -> { M , N } ~~ 2o )
12		pr2cv 7543	. . . . 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 3815	. . . . 5 |- ( M e. _V -> M e. { M , N } )
15		prid2g 3816	. . . . 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 3872	. . . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {cpr 3710   class class class wbr 4130   dom cdm 4774   ` cfv 5377   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16257  iEdgciedg 16258  UHGraphcuhgr 16312  UMGraphcumgr 16337

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16250  df-vtx 16259  df-iedg 16260  df-uhgrm 16314  df-upgren 16338  df-umgren 16339

This theorem is used by:  umgrnloop  16361  usgredgprv  16441