Theorem subumgredg2en 16148

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 4090	. 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 15990	. . . . 5 |- ( G e. UMGraph -> G e. UHGraph )
5	4	3ad2ant2 1045	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UHGraph )
6		simp1 1023	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> S SubGraph G )
7		simp3 1025	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom I )
8	2, 3, 5, 6, 7	subgruhgredgdm 16147	. . 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 2958	. . 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 2230	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
12	11	uhgrfun 15954	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
13	4, 12	syl 14	. . . . 5 |- ( G e. UMGraph -> Fun (iEdg ` G ) )
14	13	3ad2ant2 1045	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> Fun (iEdg ` G ) )
15		eqid 2230	. . . . . . 7 |- (Vtx ` S ) = (Vtx ` S )
16		eqid 2230	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
17		eqid 2230	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
18	15, 16, 3, 11, 17	subgrprop2 16137	. . . . . 6 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
19	18	simp2d 1036	. . . . 5 |- ( S SubGraph G -> I C_ (iEdg ` G ) )
20	19	3ad2ant1 1044	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> I C_ (iEdg ` G ) )
21		funssfv 5665	. . . . 5 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
22	21	eqcomd 2236	. . . 4 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
23	14, 20, 7, 22	syl3anc 1273	. . 3 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
24		simp2 1024	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UMGraph )
25	3	dmeqi 4931	. . . . . . . 8 |- dom I = dom (iEdg ` S )
26	25	eleq2i 2297	. . . . . . 7 |- ( X e. dom I <-> X e. dom (iEdg ` S ) )
27		subgreldmiedg 16146	. . . . . . . 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 1219	. . . 4 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom (iEdg ` G ) )
32	16, 11	umgredg2en 15986	. . . 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 4109	. 2 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) ~~ 2o )
35	1, 10, 34	elrabd 2963	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 1004   = wceq 1397   E.wex 1540   e. wcel 2201   {crab 2513   C_ wss 3199   ~Pcpw 3651   class class class wbr 4087   dom cdm 4724   Fun wfun 5319   ` cfv 5325   2oc2o 6578   ~~ cen 6909  Vtxcvtx 15889  iEdgciedg 15890  Edgcedg 15934  UHGraphcuhgr 15944  UMGraphcumgr 15969   SubGraph csubgr 16130

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-cnre 8145

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-iord 4462  df-on 4464  df-suc 4467  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-1o 6584  df-2o 6585  df-en 6912  df-sub 8354  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-5 9207  df-6 9208  df-7 9209  df-8 9210  df-9 9211  df-n0 9405  df-dec 9614  df-ndx 13105  df-slot 13106  df-base 13108  df-edgf 15882  df-vtx 15891  df-iedg 15892  df-edg 15935  df-uhgrm 15946  df-upgren 15970  df-umgren 15971  df-subgr 16131

This theorem is referenced by:  subumgr  16151  subusgr  16152