Theorem uhgrun 16339

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 16326	. . . . 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 2238	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		uhgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	uhgrfm 16326	. . . . . 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 2244	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3693	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2815	. . . . . 6 |- ( ph -> { s e. ~P V | E. j j e. s } = { s e. ~P (Vtx ` H ) | E. j j e. s } )
15	14	feq3d 5522	. . . . 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 5563	. . 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 4983	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4988	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2287	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		uhgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3693	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2815	. . . 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 5524	. . 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 2238	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2238	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isuhgrm 16324	. . 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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   u. cun 3218   i^i cin 3219   (/)c0 3520   ~Pcpw 3688   dom cdm 4774   -->wf 5373   ` cfv 5377  Vtxcvtx 16265  iEdgciedg 16266  UHGraphcuhgr 16320

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-dec 9780  df-ndx 13357  df-slot 13358  df-base 13360  df-edgf 16258  df-vtx 16267  df-iedg 16268  df-uhgrm 16322

This theorem is used by:  uhgrunop  16340  ushgrun  16341