Theorem upgrun 26013

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	upgrf 25981	. . . . 5 |- ( G e. UPGraph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
5	1, 4	syl 17	. . . 4 |- ( ph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
6		upgrun.h	. . . . . 6 |- ( ph -> H e. UPGraph )
7		eqid 2622	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		upgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	upgrf 25981	. . . . . 6 |- ( H e. UPGraph -> F : dom F --> { x e. ( ~P (Vtx ` H ) \ { (/) } ) | ( # ` x ) <_ 2 } )
10	6, 9	syl 17	. . . . 5 |- ( ph -> F : dom F --> { x e. ( ~P (Vtx ` H ) \ { (/) } ) | ( # ` x ) <_ 2 } )
11		upgrun.vh	. . . . . . . . . 10 |- ( ph -> (Vtx ` H ) = V )
12	11	eqcomd 2628	. . . . . . . . 9 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 4163	. . . . . . . 8 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	difeq1d 3727	. . . . . . 7 |- ( ph -> ( ~P V \ { (/) } ) = ( ~P (Vtx ` H ) \ { (/) } ) )
15	14	rabeqdv 3194	. . . . . 6 |- ( ph -> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P (Vtx ` H ) \ { (/) } ) | ( # ` x ) <_ 2 } )
16	15	feq3d 6032	. . . . 5 |- ( ph -> ( F : dom F --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> F : dom F --> { x e. ( ~P (Vtx ` H ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
17	10, 16	mpbird 247	. . . 4 |- ( ph -> F : dom F --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
18		upgrun.i	. . . 4 |- ( ph -> ( dom E i^i dom F ) = (/) )
19	5, 17, 18	fun2d 6068	. . 3 |- ( ph -> ( E u. F ) : ( dom E u. dom F ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
20		upgrun.un	. . . 4 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
21	20	dmeqd 5326	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
22		dmun 5331	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
23	21, 22	syl6eq 2672	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
24		upgrun.v	. . . . . . 7 |- ( ph -> (Vtx ` U ) = V )
25	24	pweqd 4163	. . . . . 6 |- ( ph -> ~P (Vtx ` U ) = ~P V )
26	25	difeq1d 3727	. . . . 5 |- ( ph -> ( ~P (Vtx ` U ) \ { (/) } ) = ( ~P V \ { (/) } ) )
27	26	rabeqdv 3194	. . . 4 |- ( ph -> { x e. ( ~P (Vtx ` U ) \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
28	20, 23, 27	feq123d 6034	. . 3 |- ( ph -> ( (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ( ~P (Vtx ` U ) \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( E u. F ) : ( dom E u. dom F ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
29	19, 28	mpbird 247	. 2 |- ( ph -> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ( ~P (Vtx ` U ) \ { (/) } ) | ( # ` x ) <_ 2 } )
30		upgrun.u	. . 3 |- ( ph -> U e. W )
31		eqid 2622	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
32		eqid 2622	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
33	31, 32	isupgr 25979	. . 3 |- ( U e. W -> ( U e. UPGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ( ~P (Vtx ` U ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
34	30, 33	syl 17	. 2 |- ( ph -> ( U e. UPGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { x e. ( ~P (Vtx ` U ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
35	29, 34	mpbird 247	1 |- ( ph -> U e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-upgr 25977

This theorem is referenced by:  upgrunop  26014  uspgrun  26080