Theorem umgrun 15911

Description: The union U of two multigraphs G and H with the same vertex set V is a multigraph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
umgrun.g	|- ( ph -> G e. UMGraph )
umgrun.h	|- ( ph -> H e. UMGraph )
umgrun.e	|- E = (iEdg ` G )
umgrun.f	|- F = (iEdg ` H )
umgrun.vg	|- V = (Vtx ` G )
umgrun.vh	|- ( ph -> (Vtx ` H ) = V )
umgrun.i	|- ( ph -> ( dom E i^i dom F ) = (/) )
umgrun.u	|- ( ph -> U e. W )
umgrun.v	|- ( ph -> (Vtx ` U ) = V )
umgrun.un	|- ( ph -> (iEdg ` U ) = ( E u. F ) )

Assertion

Ref	Expression
umgrun	|- ( ph -> U e. UMGraph )

Proof of Theorem umgrun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgrun.g	. . . . 5 |- ( ph -> G e. UMGraph )
2		umgrun.vg	. . . . . 6 |- V = (Vtx ` G )
3		umgrun.e	. . . . . 6 |- E = (iEdg ` G )
4	2, 3	umgrfen 15892	. . . . 5 |- ( G e. UMGraph -> E : dom E --> { x e. ~P V | x ~~ 2o } )
5	1, 4	syl 14	. . . 4 |- ( ph -> E : dom E --> { x e. ~P V | x ~~ 2o } )
6		umgrun.h	. . . . . 6 |- ( ph -> H e. UMGraph )
7		eqid 2229	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		umgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	umgrfen 15892	. . . . . 6 |- ( H e. UMGraph -> F : dom F --> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
10	6, 9	syl 14	. . . . 5 |- ( ph -> F : dom F --> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
11		umgrun.vh	. . . . . . . . 9 |- ( ph -> (Vtx ` H ) = V )
12	11	eqcomd 2235	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3654	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2793	. . . . . 6 |- ( ph -> { x e. ~P V | x ~~ 2o } = { x e. ~P (Vtx ` H ) | x ~~ 2o } )
15	14	feq3d 5458	. . . . 5 |- ( ph -> ( F : dom F --> { x e. ~P V | x ~~ 2o } <-> F : dom F --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
16	10, 15	mpbird 167	. . . 4 |- ( ph -> F : dom F --> { x e. ~P V | x ~~ 2o } )
17		umgrun.i	. . . 4 |- ( ph -> ( dom E i^i dom F ) = (/) )
18	5, 16, 17	fun2d 5495	. . 3 |- ( ph -> ( E u. F ) : ( dom E u. dom F ) --> { x e. ~P V | x ~~ 2o } )
19		umgrun.un	. . . 4 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
20	19	dmeqd 4922	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4927	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2278	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		umgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3654	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2793	. . . 4 |- ( ph -> { x e. ~P (Vtx ` U ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
26	19, 22, 25	feq123d 5460	. . 3 |- ( ph -> ( (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | x ~~ 2o } <-> ( E u. F ) : ( dom E u. dom F ) --> { x e. ~P V | x ~~ 2o } ) )
27	18, 26	mpbird 167	. 2 |- ( ph -> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | x ~~ 2o } )
28		umgrun.u	. . 3 |- ( ph -> U e. W )
29		eqid 2229	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2229	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isumgren 15890	. . 3 |- ( U e. W -> ( U e. UMGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | x ~~ 2o } ) )
32	28, 31	syl 14	. 2 |- ( ph -> ( U e. UMGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | x ~~ 2o } ) )
33	27, 32	mpbird 167	1 |- ( ph -> U e. UMGraph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   u. cun 3195   i^i cin 3196   (/)c0 3491   ~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-nul 3492  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:  umgrunop  15912  usgrun  15976