Theorem isumgren 15890

Description: The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)

Hypotheses

Ref	Expression
isumgr.v	|- V = (Vtx ` G )
isumgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
isumgren	|- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )

Distinct variable groups:   x, G   x, V

Allowed substitution hints:   U( x)   E( x)

Proof of Theorem isumgren

Dummy variables e g h v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-umgren 15879	. . 3 |- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } }
2	1	eleq2i 2296	. 2 |- ( G e. UMGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } )
3		fveq2 5623	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isumgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2280	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4922	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2233	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4921	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2278	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5623	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isumgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2280	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3654	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2793	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
15	5, 9, 14	feq123d 5460	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )
16		vtxex 15804	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2803	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5623	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15805	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2803	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5623	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
24	23	adantr 276	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
25		simpr 110	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
26	25	dmeqd 4922	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3652	. . . . . . . . 9 |- ( v = (Vtx ` h ) -> ~P v = ~P (Vtx ` h ) )
28	27	ad2antlr 489	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2793	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P (Vtx ` h ) | x ~~ 2o } )
30	25, 26, 29	feq123d 5460	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
31	22, 24, 30	sbcied2 3066	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
32	18, 19, 31	sbcied2 3066	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
33	32	cbvabv 2354	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } }
34	15, 33	elab2g 2950	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )
35	2, 34	bitrid 192	1 |- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2799   [.wsbc 3028   ~Pcpw 3649   class class class wbr 4082   dom cdm 4716   -->wf 5310   ` cfv 5314   2oc2o 6546   ~~ cen 6875  Vtxcvtx 15798  iEdgciedg 15799  UMGraphcumgr 15877

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fo 5320  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-sub 8307  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-dec 9567  df-ndx 13021  df-slot 13022  df-base 13024  df-edgf 15791  df-vtx 15800  df-iedg 15801  df-umgren 15879

This theorem is referenced by:  wrdumgren  15891  umgrfen  15892  umgr0e  15903  umgrun  15911  umgrislfupgrdom  15914  ausgrumgrien  15953  usgrumgr  15967