Theorem ushgrun 15853

Description: The union U of two (undirected) simple hypergraphs G and H with the same vertex set V is a (not necessarily simple) hypergraph with the vertex set V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

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

Assertion

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

Proof of Theorem ushgrun

Step	Hyp	Ref	Expression
1		ushgrun.g	. . 3 |- ( ph -> G e. USHGraph )
2		ushgruhgr 15845	. . 3 |- ( G e. USHGraph -> G e. UHGraph )
3	1, 2	syl 14	. 2 |- ( ph -> G e. UHGraph )
4		ushgrun.h	. . 3 |- ( ph -> H e. USHGraph )
5		ushgruhgr 15845	. . 3 |- ( H e. USHGraph -> H e. UHGraph )
6	4, 5	syl 14	. 2 |- ( ph -> H e. UHGraph )
7		ushgrun.e	. 2 |- E = (iEdg ` G )
8		ushgrun.f	. 2 |- F = (iEdg ` H )
9		ushgrun.vg	. 2 |- V = (Vtx ` G )
10		ushgrun.vh	. 2 |- ( ph -> (Vtx ` H ) = V )
11		ushgrun.i	. 2 |- ( ph -> ( dom E i^i dom F ) = (/) )
12		ushgrun.u	. 2 |- ( ph -> U e. W )
13		ushgrun.v	. 2 |- ( ph -> (Vtx ` U ) = V )
14		ushgrun.un	. 2 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	uhgrun 15851	1 |- ( ph -> U e. UHGraph )

Syntax hints:   -> wi 4   = wceq 1375   e. wcel 2180   u. cun 3175   i^i cin 3176   (/)c0 3471   dom cdm 4696   ` cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  UHGraphcuhgr 15832  USHGraphcushgr 15833

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-uhgrm 15834  df-ushgrm 15835

This theorem is referenced by: (None)