Theorem uhgrunop 15924

Description: The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If <. V , E >. and <. V , F >. are hypergraphs, then <. V , E u. F >. is a 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 Alexander van der Vekens, 27-Dec-2017.) (Revised 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 ) = (/) )

Assertion

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

Proof of Theorem uhgrunop

Step	Hyp	Ref	Expression
1		uhgrun.g	. 2 |- ( ph -> G e. UHGraph )
2		uhgrun.h	. 2 |- ( ph -> H e. UHGraph )
3		uhgrun.e	. 2 |- E = (iEdg ` G )
4		uhgrun.f	. 2 |- F = (iEdg ` H )
5		uhgrun.vg	. 2 |- V = (Vtx ` G )
6		uhgrun.vh	. 2 |- ( ph -> (Vtx ` H ) = V )
7		uhgrun.i	. 2 |- ( ph -> ( dom E i^i dom F ) = (/) )
8		vtxex 15856	. . . . 5 |- ( G e. UHGraph -> (Vtx ` G ) e. _V )
9	1, 8	syl 14	. . . 4 |- ( ph -> (Vtx ` G ) e. _V )
10	5, 9	eqeltrid 2316	. . 3 |- ( ph -> V e. _V )
11		iedgex 15857	. . . . . 6 |- ( G e. UHGraph -> (iEdg ` G ) e. _V )
12	1, 11	syl 14	. . . . 5 |- ( ph -> (iEdg ` G ) e. _V )
13	3, 12	eqeltrid 2316	. . . 4 |- ( ph -> E e. _V )
14		iedgex 15857	. . . . . 6 |- ( H e. UHGraph -> (iEdg ` H ) e. _V )
15	2, 14	syl 14	. . . . 5 |- ( ph -> (iEdg ` H ) e. _V )
16	4, 15	eqeltrid 2316	. . . 4 |- ( ph -> F e. _V )
17		unexg 4536	. . . 4 |- ( ( E e. _V /\ F e. _V ) -> ( E u. F ) e. _V )
18	13, 16, 17	syl2anc 411	. . 3 |- ( ph -> ( E u. F ) e. _V )
19		opexg 4316	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> <. V , ( E u. F ) >. e. _V )
20	10, 18, 19	syl2anc 411	. 2 |- ( ph -> <. V , ( E u. F ) >. e. _V )
21		opvtxfv 15860	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (Vtx ` <. V , ( E u. F ) >. ) = V )
22	10, 18, 21	syl2anc 411	. 2 |- ( ph -> (Vtx ` <. V , ( E u. F ) >. ) = V )
23		opiedgfv 15863	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
24	10, 18, 23	syl2anc 411	. 2 |- ( ph -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
25	1, 2, 3, 4, 5, 6, 7, 20, 22, 24	uhgrun 15923	1 |- ( ph -> <. V , ( E u. F ) >. e. UHGraph )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2800   u. cun 3196   i^i cin 3197   (/)c0 3492   <.cop 3670   dom cdm 4721   ` cfv 5322  Vtxcvtx 15850  iEdgciedg 15851  UHGraphcuhgr 15904

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-fo 5328  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-sub 8340  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-5 9193  df-6 9194  df-7 9195  df-8 9196  df-9 9197  df-n0 9391  df-dec 9600  df-ndx 13072  df-slot 13073  df-base 13075  df-edgf 15843  df-vtx 15852  df-iedg 15853  df-uhgrm 15906

This theorem is referenced by:  ushgrunop  15926