Theorem uhgrunop 15727

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 15661	. . . . 5 |- ( G e. UHGraph -> (Vtx ` G ) e. _V )
9	1, 8	syl 14	. . . 4 |- ( ph -> (Vtx ` G ) e. _V )
10	5, 9	eqeltrid 2293	. . 3 |- ( ph -> V e. _V )
11		iedgex 15662	. . . . . 6 |- ( G e. UHGraph -> (iEdg ` G ) e. _V )
12	1, 11	syl 14	. . . . 5 |- ( ph -> (iEdg ` G ) e. _V )
13	3, 12	eqeltrid 2293	. . . 4 |- ( ph -> E e. _V )
14		iedgex 15662	. . . . . 6 |- ( H e. UHGraph -> (iEdg ` H ) e. _V )
15	2, 14	syl 14	. . . . 5 |- ( ph -> (iEdg ` H ) e. _V )
16	4, 15	eqeltrid 2293	. . . 4 |- ( ph -> F e. _V )
17		unexg 4494	. . . 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 4276	. . 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 15665	. . 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 15668	. . 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 15726	1 |- ( ph -> <. V , ( E u. F ) >. e. UHGraph )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   u. cun 3165   i^i cin 3166   (/)c0 3461   <.cop 3637   dom cdm 4679   ` 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:  ushgrunop  15729