Theorem uhgraun 24884

Description: The union of two (undirected) hypergraphs (with the same vertex set): 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), analogous to umgraun 24901. (Contributed by Alexander van der Vekens, 27-Dec-2017.)

Hypotheses

Ref	Expression
uhgraun.e	|- ( ph -> E Fn A )
uhgraun.f	|- ( ph -> F Fn B )
uhgraun.i	|- ( ph -> ( A i^i B ) = (/) )
uhgraun.ge	|- ( ph -> V UHGrph E )
uhgraun.gf	|- ( ph -> V UHGrph F )

Assertion

Ref	Expression
uhgraun	|- ( ph -> V UHGrph ( E u. F ) )

Proof of Theorem uhgraun

Step	Hyp	Ref	Expression
1		uhgraun.ge	. . . . . 6 |- ( ph -> V UHGrph E )
2		uhgraf 24872	. . . . . 6 |- ( V UHGrph E -> E : dom E --> ( ~P V \ { (/) } ) )
3	1, 2	syl 17	. . . . 5 |- ( ph -> E : dom E --> ( ~P V \ { (/) } ) )
4		uhgraun.e	. . . . 5 |- ( ph -> E Fn A )
5		fndm 5693	. . . . . . 7 |- ( E Fn A -> dom E = A )
6	5	feq2d 5733	. . . . . 6 |- ( E Fn A -> ( E : dom E --> ( ~P V \ { (/) } ) <-> E : A --> ( ~P V \ { (/) } ) ) )
7	6	biimpac 488	. . . . 5 |- ( ( E : dom E --> ( ~P V \ { (/) } ) /\ E Fn A ) -> E : A --> ( ~P V \ { (/) } ) )
8	3, 4, 7	syl2anc 665	. . . 4 |- ( ph -> E : A --> ( ~P V \ { (/) } ) )
9		uhgraun.gf	. . . . . 6 |- ( ph -> V UHGrph F )
10		uhgraf 24872	. . . . . 6 |- ( V UHGrph F -> F : dom F --> ( ~P V \ { (/) } ) )
11	9, 10	syl 17	. . . . 5 |- ( ph -> F : dom F --> ( ~P V \ { (/) } ) )
12		uhgraun.f	. . . . 5 |- ( ph -> F Fn B )
13		fndm 5693	. . . . . . 7 |- ( F Fn B -> dom F = B )
14	13	feq2d 5733	. . . . . 6 |- ( F Fn B -> ( F : dom F --> ( ~P V \ { (/) } ) <-> F : B --> ( ~P V \ { (/) } ) ) )
15	14	biimpac 488	. . . . 5 |- ( ( F : dom F --> ( ~P V \ { (/) } ) /\ F Fn B ) -> F : B --> ( ~P V \ { (/) } ) )
16	11, 12, 15	syl2anc 665	. . . 4 |- ( ph -> F : B --> ( ~P V \ { (/) } ) )
17		uhgraun.i	. . . 4 |- ( ph -> ( A i^i B ) = (/) )
18		fun2 5764	. . . 4 |- ( ( ( E : A --> ( ~P V \ { (/) } ) /\ F : B --> ( ~P V \ { (/) } ) ) /\ ( A i^i B ) = (/) ) -> ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) )
19	8, 16, 17, 18	syl21anc 1263	. . 3 |- ( ph -> ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) )
20		fdm 5750	. . . . 5 |- ( ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) -> dom ( E u. F ) = ( A u. B ) )
21	19, 20	syl 17	. . . 4 |- ( ph -> dom ( E u. F ) = ( A u. B ) )
22	21	feq2d 5733	. . 3 |- ( ph -> ( ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) <-> ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) ) )
23	19, 22	mpbird 235	. 2 |- ( ph -> ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) )
24		reluhgra 24867	. . . 4 |- Rel UHGrph
25		releldm 5087	. . . 4 |- ( ( Rel UHGrph /\ V UHGrph E ) -> V e. dom UHGrph )
26	24, 1, 25	sylancr 667	. . 3 |- ( ph -> V e. dom UHGrph )
27		uhgrav 24869	. . . . . 6 |- ( V UHGrph E -> ( V e. _V /\ E e. _V ) )
28	27	simprd 464	. . . . 5 |- ( V UHGrph E -> E e. _V )
29	1, 28	syl 17	. . . 4 |- ( ph -> E e. _V )
30		uhgrav 24869	. . . . . 6 |- ( V UHGrph F -> ( V e. _V /\ F e. _V ) )
31	30	simprd 464	. . . . 5 |- ( V UHGrph F -> F e. _V )
32	9, 31	syl 17	. . . 4 |- ( ph -> F e. _V )
33		unexg 6606	. . . 4 |- ( ( E e. _V /\ F e. _V ) -> ( E u. F ) e. _V )
34	29, 32, 33	syl2anc 665	. . 3 |- ( ph -> ( E u. F ) e. _V )
35		isuhgra 24871	. . 3 |- ( ( V e. dom UHGrph /\ ( E u. F ) e. _V ) -> ( V UHGrph ( E u. F ) <-> ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) ) )
36	26, 34, 35	syl2anc 665	. 2 |- ( ph -> ( V UHGrph ( E u. F ) <-> ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) ) )
37	23, 36	mpbird 235	1 |- ( ph -> V UHGrph ( E u. F ) )

Syntax hints:   -> wi 4   <-> wb 187   = wceq 1437   e. wcel 1870   _Vcvv 3087   \ cdif 3439   u. cun 3440   i^i cin 3441   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426   dom cdm 4854   Rel wrel 4859   Fn wfn 5596   -->wf 5597   UHGrph cuhg 24863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-fun 5603  df-fn 5604  df-f 5605  df-uhgra 24865

This theorem is referenced by: (None)