Theorem subumgredg2en 16426

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 4128	. 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 16268	. . . . 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 16425	. . 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 16232	. . . . . 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 16415	. . . . . 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 5716	. . . . 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 4977	. . . . . . . 8 |- dom I = dom (iEdg ` S )
26	25	eleq2i 2305	. . . . . . 7 |- ( X e. dom I <-> X e. dom (iEdg ` S ) )
27		subgreldmiedg 16424	. . . . . . . 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 16264	. . . 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 4147	. 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 } )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   Fun wfun 5366   ` cfv 5372   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UHGraphcuhgr 16222  UMGraphcumgr 16247   SubGraph csubgr 16408

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-upgren 16248  df-umgren 16249  df-subgr 16409

This theorem is referenced by:  subumgr  16429  subusgr  16430