Theorem isumgren 16092

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 16081	. . 3 |- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } }
2	1	eleq2i 2299	. 2 |- ( G e. UMGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } )
3		fveq2 5669	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isumgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2283	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4957	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2236	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4956	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2281	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5669	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isumgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2283	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3673	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2806	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
15	5, 9, 14	feq123d 5498	. . 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 16005	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2816	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5669	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 16006	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2816	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5669	. . . . . . 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 4957	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3671	. . . . . . . . 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 2806	. . . . . . 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 5498	. . . . . 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 3079	. . . . 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 3079	. . . 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 2359	. . 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 2963	. 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 1398   e. wcel 2203   {cab 2218   {crab 2524   _Vcvv 2812   [.wsbc 3041   ~Pcpw 3668   class class class wbr 4108   dom cdm 4748   -->wf 5347   ` cfv 5351   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15999  iEdgciedg 16000  UMGraphcumgr 16079

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8445  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-dec 9709  df-ndx 13207  df-slot 13208  df-base 13210  df-edgf 15992  df-vtx 16001  df-iedg 16002  df-umgren 16081

This theorem is referenced by:  wrdumgren  16093  umgrfen  16094  umgr0e  16105  umgr1een  16112  umgrun  16115  umgrislfupgrdom  16118  ausgrumgrien  16157  usgrumgr  16171  subumgr  16261