Theorem uhgrun 15726

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 15713	. . . . 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 2206	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
8		uhgrun.f	. . . . . . 7 |- F = (iEdg ` H )
9	7, 8	uhgrfm 15713	. . . . . 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 2212	. . . . . . . 8 |- ( ph -> V = (Vtx ` H ) )
13	12	pweqd 3622	. . . . . . 7 |- ( ph -> ~P V = ~P (Vtx ` H ) )
14	13	rabeqdv 2767	. . . . . 6 |- ( ph -> { s e. ~P V | E. j j e. s } = { s e. ~P (Vtx ` H ) | E. j j e. s } )
15	14	feq3d 5420	. . . . 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 5456	. . 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 4885	. . . . 5 |- ( ph -> dom (iEdg ` U ) = dom ( E u. F ) )
21		dmun 4890	. . . . 5 |- dom ( E u. F ) = ( dom E u. dom F )
22	20, 21	eqtrdi 2255	. . . 4 |- ( ph -> dom (iEdg ` U ) = ( dom E u. dom F ) )
23		uhgrun.v	. . . . . 6 |- ( ph -> (Vtx ` U ) = V )
24	23	pweqd 3622	. . . . 5 |- ( ph -> ~P (Vtx ` U ) = ~P V )
25	24	rabeqdv 2767	. . . 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 5422	. . 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 2206	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
30		eqid 2206	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
31	29, 30	isuhgrm 15711	. . 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 1373   E.wex 1516   e. wcel 2177   {crab 2489   u. cun 3165   i^i cin 3166   (/)c0 3461   ~Pcpw 3617   dom cdm 4679   -->wf 5272   ` cfv 5276  Vtxcvtx 15655  iEdgciedg 15656  UHGraphcuhgr 15707

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fo 5282  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-sub 8252  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-dec 9512  df-ndx 12879  df-slot 12880  df-base 12882  df-edgf 15648  df-vtx 15657  df-iedg 15658  df-uhgrm 15709

This theorem is referenced by:  uhgrunop  15727  ushgrun  15728