Theorem uhgraun 24125

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 24142. (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 24113	. . . . . 6 |- ( V UHGrph E -> E : dom E --> ( ~P V \ { (/) } ) )
3	1, 2	syl 16	. . . . 5 |- ( ph -> E : dom E --> ( ~P V \ { (/) } ) )
4		uhgraun.e	. . . . 5 |- ( ph -> E Fn A )
5		fndm 5686	. . . . . . 7 |- ( E Fn A -> dom E = A )
6	5	feq2d 5724	. . . . . 6 |- ( E Fn A -> ( E : dom E --> ( ~P V \ { (/) } ) <-> E : A --> ( ~P V \ { (/) } ) ) )
7	6	biimpac 486	. . . . 5 |- ( ( E : dom E --> ( ~P V \ { (/) } ) /\ E Fn A ) -> E : A --> ( ~P V \ { (/) } ) )
8	3, 4, 7	syl2anc 661	. . . 4 |- ( ph -> E : A --> ( ~P V \ { (/) } ) )
9		uhgraun.gf	. . . . . 6 |- ( ph -> V UHGrph F )
10		uhgraf 24113	. . . . . 6 |- ( V UHGrph F -> F : dom F --> ( ~P V \ { (/) } ) )
11	9, 10	syl 16	. . . . 5 |- ( ph -> F : dom F --> ( ~P V \ { (/) } ) )
12		uhgraun.f	. . . . 5 |- ( ph -> F Fn B )
13		fndm 5686	. . . . . . 7 |- ( F Fn B -> dom F = B )
14	13	feq2d 5724	. . . . . 6 |- ( F Fn B -> ( F : dom F --> ( ~P V \ { (/) } ) <-> F : B --> ( ~P V \ { (/) } ) ) )
15	14	biimpac 486	. . . . 5 |- ( ( F : dom F --> ( ~P V \ { (/) } ) /\ F Fn B ) -> F : B --> ( ~P V \ { (/) } ) )
16	11, 12, 15	syl2anc 661	. . . 4 |- ( ph -> F : B --> ( ~P V \ { (/) } ) )
17		uhgraun.i	. . . 4 |- ( ph -> ( A i^i B ) = (/) )
18		fun2 5755	. . . 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 1227	. . 3 |- ( ph -> ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) )
20		fdm 5741	. . . . 5 |- ( ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) -> dom ( E u. F ) = ( A u. B ) )
21	19, 20	syl 16	. . . 4 |- ( ph -> dom ( E u. F ) = ( A u. B ) )
22	21	feq2d 5724	. . 3 |- ( ph -> ( ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) <-> ( E u. F ) : ( A u. B ) --> ( ~P V \ { (/) } ) ) )
23	19, 22	mpbird 232	. 2 |- ( ph -> ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) )
24		reluhgra 24108	. . . 4 |- Rel UHGrph
25		releldm 5241	. . . 4 |- ( ( Rel UHGrph /\ V UHGrph E ) -> V e. dom UHGrph )
26	24, 1, 25	sylancr 663	. . 3 |- ( ph -> V e. dom UHGrph )
27		uhgrav 24110	. . . . . 6 |- ( V UHGrph E -> ( V e. _V /\ E e. _V ) )
28	27	simprd 463	. . . . 5 |- ( V UHGrph E -> E e. _V )
29	1, 28	syl 16	. . . 4 |- ( ph -> E e. _V )
30		uhgrav 24110	. . . . . 6 |- ( V UHGrph F -> ( V e. _V /\ F e. _V ) )
31	30	simprd 463	. . . . 5 |- ( V UHGrph F -> F e. _V )
32	9, 31	syl 16	. . . 4 |- ( ph -> F e. _V )
33		unexg 6596	. . . 4 |- ( ( E e. _V /\ F e. _V ) -> ( E u. F ) e. _V )
34	29, 32, 33	syl2anc 661	. . 3 |- ( ph -> ( E u. F ) e. _V )
35		isuhgra 24112	. . 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 661	. 2 |- ( ph -> ( V UHGrph ( E u. F ) <-> ( E u. F ) : dom ( E u. F ) --> ( ~P V \ { (/) } ) ) )
37	23, 36	mpbird 232	1 |- ( ph -> V UHGrph ( E u. F ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   _Vcvv 3118   \ cdif 3478   u. cun 3479   i^i cin 3480   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453   dom cdm 5005   Rel wrel 5010   Fn wfn 5589   -->wf 5590   UHGrph cuhg 24104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-fun 5596  df-fn 5597  df-f 5598  df-uhgra 24106

This theorem is referenced by: (None)