Theorem umgrun 16235

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 16214	. . . . 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 2234	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		umgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	umgrfen 16214	. . . . . 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 2240	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3679	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2809	. . . . . 6 |- ( ph -> { x e. ~P V | x ~~ 2o } = { x e. ~P (Vtx ` H ) | x ~~ 2o } )
15	14	feq3d 5502	. . . . 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 5543	. . 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 4963	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4968	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2283	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		umgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3679	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2809	. . . 4 |- ( ph -> { x e. ~P (Vtx ` U ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
26	19, 22, 25	feq123d 5504	. . 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 2234	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2234	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isumgren 16212	. . 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 1398   e. wcel 2205   {crab 2526   u. cun 3212   i^i cin 3213   (/)c0 3512   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   -->wf 5353   ` cfv 5357   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16119  iEdgciedg 16120  UMGraphcumgr 16199

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16112  df-vtx 16121  df-iedg 16122  df-umgren 16201

This theorem is referenced by:  umgrunop  16236  usgrun  16300