Theorem ushgrunop 16330

Description: The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If <. V , E >. and <. V , F >. are simple hypergraphs, then <. V , E u. F >. is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (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 ) = (/) )

Assertion

Ref	Expression
ushgrunop	|- ( ph -> <. V , ( E u. F ) >. e. UHGraph )

Proof of Theorem ushgrunop

Step	Hyp	Ref	Expression
1		ushgrun.g	. . 3 |- ( ph -> G e. USHGraph )
2		ushgruhgr 16321	. . 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 16321	. . 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	3, 6, 7, 8, 9, 10, 11	uhgrunop 16328	1 |- ( ph -> <. V , ( E u. F ) >. e. UHGraph )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   u. cun 3218   i^i cin 3219   (/)c0 3520   <.cop 3712   dom cdm 4774   ` cfv 5377  Vtxcvtx 16253  iEdgciedg 16254  UHGraphcuhgr 16308  USHGraphcushgr 16309

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-f1 5382  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 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-uhgrm 16310  df-ushgrm 16311

This theorem is used by: (None)