Theorem upgrun 16047

Description: The union U of two pseudographs G and H with the same vertex set V is a pseudograph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
upgrun	|- ( ph -> U e. UPGraph )

Proof of Theorem upgrun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrun.g	. . . . 5 |- ( ph -> G e. UPGraph )
2		upgrun.vg	. . . . . 6 |- V = (Vtx ` G )
3		upgrun.e	. . . . . 6 |- E = (iEdg ` G )
4	2, 3	upgrfen 16018	. . . . 5 |- ( G e. UPGraph -> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	1, 4	syl 14	. . . 4 |- ( ph -> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6		upgrun.h	. . . . . 6 |- ( ph -> H e. UPGraph )
7		eqid 2231	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		upgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	upgrfen 16018	. . . . . 6 |- ( H e. UPGraph -> F : dom F --> { x e. ~P (Vtx ` H ) | ( x ~~ 1o \/ x ~~ 2o ) } )
10	6, 9	syl 14	. . . . 5 |- ( ph -> F : dom F --> { x e. ~P (Vtx ` H ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11		upgrun.vh	. . . . . . . . 9 |- ( ph -> (Vtx ` H ) = V )
12	11	eqcomd 2237	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3661	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2797	. . . . . 6 |- ( ph -> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P (Vtx ` H ) | ( x ~~ 1o \/ x ~~ 2o ) } )
15	14	feq3d 5478	. . . . 5 |- ( ph -> ( F : dom F --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> F : dom F --> { x e. ~P (Vtx ` H ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
16	10, 15	mpbird 167	. . . 4 |- ( ph -> F : dom F --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
17		upgrun.i	. . . 4 |- ( ph -> ( dom E i^i dom F ) = (/) )
18	5, 16, 17	fun2d 5518	. . 3 |- ( ph -> ( E u. F ) : ( dom E u. dom F ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
19		upgrun.un	. . . 4 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
20	19	dmeqd 4939	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4944	. . . . 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		upgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3661	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2797	. . . 4 |- ( ph -> { x e. ~P (Vtx ` U ) | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
26	19, 22, 25	feq123d 5480	. . 3 |- ( ph -> ( (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( E u. F ) : ( dom E u. dom F ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
27	18, 26	mpbird 167	. 2 |- ( ph -> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | ( x ~~ 1o \/ x ~~ 2o ) } )
28		upgrun.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	isupgren 16016	. . 3 |- ( U e. W -> ( U e. UPGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
32	28, 31	syl 14	. 2 |- ( ph -> ( U e. UPGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ~P (Vtx ` U ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
33	27, 32	mpbird 167	1 |- ( ph -> U e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   {crab 2515   u. cun 3199   i^i cin 3200   (/)c0 3496   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   -->wf 5329   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15933  iEdgciedg 15934  UPGraphcupgr 16012

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8395  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-dec 9655  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15926  df-vtx 15935  df-iedg 15936  df-upgren 16014

This theorem is referenced by:  upgrunop  16048  uspgrun  16112  vtxdfifiun  16218