Theorem subumgredg2en 16121

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 4091	. 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 15963	. . . . 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 16120	. . 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 2959	. . 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 2231	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
12	11	uhgrfun 15927	. . . . . 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 2231	. . . . . . 7 |- (Vtx ` S ) = (Vtx ` S )
16		eqid 2231	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
17		eqid 2231	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
18	15, 16, 3, 11, 17	subgrprop2 16110	. . . . . 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 2237	. . . 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 4932	. . . . . . . 8 |- dom I = dom (iEdg ` S )
26	25	eleq2i 2298	. . . . . . 7 |- ( X e. dom I <-> X e. dom (iEdg ` S ) )
27		subgreldmiedg 16119	. . . . . . . 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 15959	. . . 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 4110	. 2 |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) ~~ 2o )
35	1, 10, 34	elrabd 2964	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 2202   {crab 2514   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   Fun wfun 5320   ` cfv 5326   2oc2o 6575   ~~ cen 6906  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UHGraphcuhgr 15917  UMGraphcumgr 15942   SubGraph csubgr 16103

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-en 6909  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919  df-upgren 15943  df-umgren 15944  df-subgr 16104

This theorem is referenced by:  subumgr  16124  subusgr  16125