Theorem uhgreq12g 25960

Description: If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)

Hypotheses

Ref	Expression
uhgrf.v	|- V = (Vtx ` G )
uhgrf.e	|- E = (iEdg ` G )
uhgreq12g.w	|- W = (Vtx ` H )
uhgreq12g.f	|- F = (iEdg ` H )

Assertion

Ref	Expression
uhgreq12g	|- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( G e. UHGraph <-> H e. UHGraph ) )

Proof of Theorem uhgreq12g

Step	Hyp	Ref	Expression
1		uhgrf.v	. . . . 5 |- V = (Vtx ` G )
2		uhgrf.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	isuhgr 25955	. . . 4 |- ( G e. X -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )
4	3	adantr 481	. . 3 |- ( ( G e. X /\ H e. Y ) -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )
5	4	adantr 481	. 2 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )
6		simpr 477	. . . 4 |- ( ( V = W /\ E = F ) -> E = F )
7	6	dmeqd 5326	. . . 4 |- ( ( V = W /\ E = F ) -> dom E = dom F )
8		pweq 4161	. . . . . 6 |- ( V = W -> ~P V = ~P W )
9	8	difeq1d 3727	. . . . 5 |- ( V = W -> ( ~P V \ { (/) } ) = ( ~P W \ { (/) } ) )
10	9	adantr 481	. . . 4 |- ( ( V = W /\ E = F ) -> ( ~P V \ { (/) } ) = ( ~P W \ { (/) } ) )
11	6, 7, 10	feq123d 6034	. . 3 |- ( ( V = W /\ E = F ) -> ( E : dom E --> ( ~P V \ { (/) } ) <-> F : dom F --> ( ~P W \ { (/) } ) ) )
12		uhgreq12g.w	. . . . . 6 |- W = (Vtx ` H )
13		uhgreq12g.f	. . . . . 6 |- F = (iEdg ` H )
14	12, 13	isuhgr 25955	. . . . 5 |- ( H e. Y -> ( H e. UHGraph <-> F : dom F --> ( ~P W \ { (/) } ) ) )
15	14	adantl 482	. . . 4 |- ( ( G e. X /\ H e. Y ) -> ( H e. UHGraph <-> F : dom F --> ( ~P W \ { (/) } ) ) )
16	15	bicomd 213	. . 3 |- ( ( G e. X /\ H e. Y ) -> ( F : dom F --> ( ~P W \ { (/) } ) <-> H e. UHGraph ) )
17	11, 16	sylan9bbr 737	. 2 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( E : dom E --> ( ~P V \ { (/) } ) <-> H e. UHGraph ) )
18	5, 17	bitrd 268	1 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( G e. UHGraph <-> H e. UHGraph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   -->wf 5884   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-uhgr 25953

This theorem is referenced by: (None)