Theorem subumgredg2en 16392

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 4117	. 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 16234	. . . . 5 |- ( G e. UMGraph -> G e. UHGraph )
5	4	3ad2ant2 1046	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UHGraph )
6		simp1 1024	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> S SubGraph G )
7		simp3 1026	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom I )
8	2, 3, 5, 6, 7	subgruhgredgdm 16391	. . 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 2973	. . 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 2234	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
12	11	uhgrfun 16198	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
13	4, 12	syl 14	. . . . 5 |- ( G e. UMGraph -> Fun (iEdg ` G ) )
14	13	3ad2ant2 1046	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> Fun (iEdg ` G ) )
15		eqid 2234	. . . . . . 7 |- (Vtx ` S ) = (Vtx ` S )
16		eqid 2234	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
17		eqid 2234	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
18	15, 16, 3, 11, 17	subgrprop2 16381	. . . . . 6 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
19	18	simp2d 1037	. . . . 5 |- ( S SubGraph G -> I C_ (iEdg ` G ) )
20	19	3ad2ant1 1045	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> I C_ (iEdg ` G ) )
21		funssfv 5701	. . . . 5 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
22	21	eqcomd 2240	. . . 4 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
23	14, 20, 7, 22	syl3anc 1274	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
24		simp2 1025	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UMGraph )
25	3	dmeqi 4962	. . . . . . . 8 |- dom I = dom (iEdg ` S )
26	25	eleq2i 2301	. . . . . . 7 |- ( X e. dom I <-> X e. dom (iEdg ` S ) )
27		subgreldmiedg 16390	. . . . . . . 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 1220	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom (iEdg ` G ) )
32	16, 11	umgredg2en 16230	. . . 4 |- ( ( G e. UMGraph /\ X e. dom (iEdg ` G ) ) -> ( (iEdg ` G ) ` X ) ~~ 2o )
33	24, 31, 32	syl2anc 411	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) ~~ 2o )
34	23, 33	eqbrtrd 4136	. 2 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) ~~ 2o )
35	1, 10, 34	elrabd 2978	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 1005   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   Fun wfun 5351   ` cfv 5357   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178  UHGraphcuhgr 16188  UMGraphcumgr 16213   SubGraph csubgr 16374

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 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-upgren 16214  df-umgren 16215  df-subgr 16375

This theorem is referenced by:  subumgr  16395  subusgr  16396