Theorem uhgrun 15930

Description: The union U of two (undirected) hypergraphs G and H with the same vertex set V is a hypergraph with the vertex set V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
uhgrun	|- ( ph -> U e. UHGraph )

Proof of Theorem uhgrun

Dummy variables s j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrun.g	. . . . 5 |- ( ph -> G e. UHGraph )
2		uhgrun.vg	. . . . . 6 |- V = (Vtx ` G )
3		uhgrun.e	. . . . . 6 |- E = (iEdg ` G )
4	2, 3	uhgrfm 15917	. . . . 5 |- ( G e. UHGraph -> E : dom E --> { s e. ~P V | E. j j e. s } )
5	1, 4	syl 14	. . . 4 |- ( ph -> E : dom E --> { s e. ~P V | E. j j e. s } )
6		uhgrun.h	. . . . . 6 |- ( ph -> H e. UHGraph )
7		eqid 2229	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		uhgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	uhgrfm 15917	. . . . . 6 |- ( H e. UHGraph -> F : dom F --> { s e. ~P (Vtx ` H ) | E. j j e. s } )
10	6, 9	syl 14	. . . . 5 |- ( ph -> F : dom F --> { s e. ~P (Vtx ` H ) | E. j j e. s } )
11		uhgrun.vh	. . . . . . . . 9 |- ( ph -> (Vtx ` H ) = V )
12	11	eqcomd 2235	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3655	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2794	. . . . . 6 |- ( ph -> { s e. ~P V | E. j j e. s } = { s e. ~P (Vtx ` H ) | E. j j e. s } )
15	14	feq3d 5468	. . . . 5 |- ( ph -> ( F : dom F --> { s e. ~P V | E. j j e. s } <-> F : dom F --> { s e. ~P (Vtx ` H ) | E. j j e. s } ) )
16	10, 15	mpbird 167	. . . 4 |- ( ph -> F : dom F --> { s e. ~P V | E. j j e. s } )
17		uhgrun.i	. . . 4 |- ( ph -> ( dom E i^i dom F ) = (/) )
18	5, 16, 17	fun2d 5507	. . 3 |- ( ph -> ( E u. F ) : ( dom E u. dom F ) --> { s e. ~P V | E. j j e. s } )
19		uhgrun.un	. . . 4 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
20	19	dmeqd 4931	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4936	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2278	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		uhgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3655	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2794	. . . 4 |- ( ph -> { s e. ~P (Vtx ` U ) | E. j j e. s } = { s e. ~P V | E. j j e. s } )
26	19, 22, 25	feq123d 5470	. . 3 |- ( ph -> ( (iEdg ` U ) : dom (iEdg ` U ) --> { s e. ~P (Vtx ` U ) | E. j j e. s } <-> ( E u. F ) : ( dom E u. dom F ) --> { s e. ~P V | E. j j e. s } ) )
27	18, 26	mpbird 167	. 2 |- ( ph -> (iEdg ` U ) : dom (iEdg ` U ) --> { s e. ~P (Vtx ` U ) | E. j j e. s } )
28		uhgrun.u	. . 3 |- ( ph -> U e. W )
29		eqid 2229	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2229	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isuhgrm 15915	. . 3 |- ( U e. W -> ( U e. UHGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { s e. ~P (Vtx ` U ) | E. j j e. s } ) )
32	28, 31	syl 14	. 2 |- ( ph -> ( U e. UHGraph <-> (iEdg ` U ) : dom (iEdg ` U ) --> { s e. ~P (Vtx ` U ) | E. j j e. s } ) )
33	27, 32	mpbird 167	1 |- ( ph -> U e. UHGraph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {crab 2512   u. cun 3196   i^i cin 3197   (/)c0 3492   ~Pcpw 3650   dom cdm 4723   -->wf 5320   ` cfv 5324  Vtxcvtx 15856  iEdgciedg 15857  UHGraphcuhgr 15911

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-uhgrm 15913

This theorem is referenced by:  uhgrunop  15931  ushgrun  15932