Theorem umgrun 15985

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 15964	. . . . 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 2231	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		umgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	umgrfen 15964	. . . . . 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 2237	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3657	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2796	. . . . . 6 |- ( ph -> { x e. ~P V | x ~~ 2o } = { x e. ~P (Vtx ` H ) | x ~~ 2o } )
15	14	feq3d 5471	. . . . 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 5510	. . 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 4933	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4938	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2280	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		umgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3657	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2796	. . . 4 |- ( ph -> { x e. ~P (Vtx ` U ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
26	19, 22, 25	feq123d 5473	. . 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 2231	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2231	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isumgren 15962	. . 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 1397   e. wcel 2202   {crab 2514   u. cun 3198   i^i cin 3199   (/)c0 3494   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   -->wf 5322   ` cfv 5326   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15869  iEdgciedg 15870  UMGraphcumgr 15949

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-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

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-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872  df-umgren 15951

This theorem is referenced by:  umgrunop  15986  usgrun  16050