Theorem uhgrun 15966

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 15953	. . . . 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 2230	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		uhgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	uhgrfm 15953	. . . . . 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 2236	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3658	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2795	. . . . . 6 |- ( ph -> { s e. ~P V | E. j j e. s } = { s e. ~P (Vtx ` H ) | E. j j e. s } )
15	14	feq3d 5473	. . . . 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 5512	. . 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 4935	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4940	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2279	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		uhgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3658	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2795	. . . 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 5475	. . 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 2230	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2230	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isuhgrm 15951	. . 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 1397   E.wex 1540   e. wcel 2201   {crab 2513   u. cun 3197   i^i cin 3198   (/)c0 3493   ~Pcpw 3653   dom cdm 4727   -->wf 5324   ` cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  UHGraphcuhgr 15947

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fo 5334  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-uhgrm 15949

This theorem is referenced by:  uhgrunop  15967  ushgrun  15968