Theorem subumgredg2en 16512

Description: An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.)

Hypotheses

Ref	Expression
subumgredg2.v	|- V = (Vtx ` S )
subumgredg2.i	|- I = (iEdg ` S )

Assertion

Ref	Expression
subumgredg2en	|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { e e. ~P V | e ~~ 2o } )

Distinct variable groups:   e, I   e, V   e, X

Allowed substitution hints:   S( e)   G( e)

Proof of Theorem subumgredg2en

Dummy variables j s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4133	. 2 |- ( e = ( I ` X ) -> ( e ~~ 2o <-> ( I ` X ) ~~ 2o ) )
2		subumgredg2.v	. . . 4 |- V = (Vtx ` S )
3		subumgredg2.i	. . . 4 |- I = (iEdg ` S )
4		umgruhgr 16354	. . . . 5 |- ( G e. UMGraph -> G e. UHGraph )
5	4	3ad2ant2 1050	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UHGraph )
6		simp1 1028	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> S SubGraph G )
7		simp3 1030	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom I )
8	2, 3, 5, 6, 7	subgruhgredgdm 16511	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )
9		elrabi 2979	. . 3 |- ( ( I ` X ) e. { s e. ~P V | E. j j e. s } -> ( I ` X ) e. ~P V )
10	8, 9	syl 14	. 2 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. ~P V )
11		eqid 2238	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
12	11	uhgrfun 16318	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
13	4, 12	syl 14	. . . . 5 |- ( G e. UMGraph -> Fun (iEdg ` G ) )
14	13	3ad2ant2 1050	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> Fun (iEdg ` G ) )
15		eqid 2238	. . . . . . 7 |- (Vtx ` S ) = (Vtx ` S )
16		eqid 2238	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
17		eqid 2238	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
18	15, 16, 3, 11, 17	subgrprop2 16501	. . . . . 6 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
19	18	simp2d 1041	. . . . 5 |- ( S SubGraph G -> I C_ (iEdg ` G ) )
20	19	3ad2ant1 1049	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> I C_ (iEdg ` G ) )
21		funssfv 5721	. . . . 5 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
22	21	eqcomd 2244	. . . 4 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
23	14, 20, 7, 22	syl3anc 1278	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
24		simp2 1029	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UMGraph )
25	3	dmeqi 4982	. . . . . . . 8 |- dom I = dom (iEdg ` S )
26	25	eleq2i 2305	. . . . . . 7 |- ( X e. dom I <-> X e. dom (iEdg ` S ) )
27		subgreldmiedg 16510	. . . . . . . 8 |- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )
28	27	ex 115	. . . . . . 7 |- ( S SubGraph G -> ( X e. dom (iEdg ` S ) -> X e. dom (iEdg ` G ) ) )
29	26, 28	biimtrid 152	. . . . . 6 |- ( S SubGraph G -> ( X e. dom I -> X e. dom (iEdg ` G ) ) )
30	29	a1d 22	. . . . 5 |- ( S SubGraph G -> ( G e. UMGraph -> ( X e. dom I -> X e. dom (iEdg ` G ) ) ) )
31	30	3imp 1224	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom (iEdg ` G ) )
32	16, 11	umgredg2en 16350	. . . 4 |- ( ( G e. UMGraph /\ X e. dom (iEdg ` G ) ) -> ( (iEdg ` G ) ` X ) ~~ 2o )
33	24, 31, 32	syl2anc 415	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) ~~ 2o )
34	23, 33	eqbrtrd 4152	. 2 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) ~~ 2o )
35	1, 10, 34	elrabd 2984	1 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { e e. ~P V | e ~~ 2o } )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   dom cdm 4774   Fun wfun 5371   ` cfv 5377   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UHGraphcuhgr 16308  UMGraphcumgr 16333   SubGraph csubgr 16494

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-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 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-umgren 16335  df-subgr 16495

This theorem is used by:  subumgr  16515  subusgr  16516