Theorem uhgrun 16130

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 16117	. . . . 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 2234	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		uhgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	uhgrfm 16117	. . . . . 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 2240	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3676	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2809	. . . . . 6 |- ( ph -> { s e. ~P V | E. j j e. s } = { s e. ~P (Vtx ` H ) | E. j j e. s } )
15	14	feq3d 5499	. . . . 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 5540	. . 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 4960	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4965	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2283	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		uhgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3676	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2809	. . . 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 5501	. . 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 2234	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2234	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isuhgrm 16115	. . 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 1398   E.wex 1541   e. wcel 2205   {crab 2526   u. cun 3211   i^i cin 3212   (/)c0 3510   ~Pcpw 3671   dom cdm 4751   -->wf 5350   ` cfv 5354  Vtxcvtx 16056  iEdgciedg 16057  UHGraphcuhgr 16111

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-uhgrm 16113

This theorem is referenced by:  uhgrunop  16131  ushgrun  16132