Theorem umgrun 15769

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 15753	. . . . 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 2206	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		umgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	umgrfen 15753	. . . . . 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 2212	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3623	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2767	. . . . . 6 |- ( ph -> { x e. ~P V | x ~~ 2o } = { x e. ~P (Vtx ` H ) | x ~~ 2o } )
15	14	feq3d 5421	. . . . 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 5458	. . 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 4886	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4891	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2255	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		umgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3623	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2767	. . . 4 |- ( ph -> { x e. ~P (Vtx ` U ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
26	19, 22, 25	feq123d 5423	. . 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 2206	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2206	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isumgren 15751	. . 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 1373   e. wcel 2177   {crab 2489   u. cun 3166   i^i cin 3167   (/)c0 3462   ~Pcpw 3618   class class class wbr 4048   dom cdm 4680   -->wf 5273   ` cfv 5277   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UMGraphcumgr 15738

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fo 5283  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-umgren 15740

This theorem is referenced by:  umgrunop  15770