Theorem uhgreq12g 15841

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

Dummy variables s j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrf.v	. . . . 5 |- V = (Vtx ` G )
2		uhgrf.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	isuhgrm 15836	. . . 4 |- ( G e. X -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
4	3	adantr 276	. . 3 |- ( ( G e. X /\ H e. Y ) -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
5	4	adantr 276	. 2 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
6		simpr 110	. . . 4 |- ( ( V = W /\ E = F ) -> E = F )
7	6	dmeqd 4902	. . . 4 |- ( ( V = W /\ E = F ) -> dom E = dom F )
8		pweq 3632	. . . . . 6 |- ( V = W -> ~P V = ~P W )
9	8	rabeqdv 2773	. . . . 5 |- ( V = W -> { s e. ~P V | E. j j e. s } = { s e. ~P W | E. j j e. s } )
10	9	adantr 276	. . . 4 |- ( ( V = W /\ E = F ) -> { s e. ~P V | E. j j e. s } = { s e. ~P W | E. j j e. s } )
11	6, 7, 10	feq123d 5440	. . 3 |- ( ( V = W /\ E = F ) -> ( E : dom E --> { s e. ~P V | E. j j e. s } <-> F : dom F --> { s e. ~P W | E. j j e. s } ) )
12		uhgreq12g.w	. . . . . 6 |- W = (Vtx ` H )
13		uhgreq12g.f	. . . . . 6 |- F = (iEdg ` H )
14	12, 13	isuhgrm 15836	. . . . 5 |- ( H e. Y -> ( H e. UHGraph <-> F : dom F --> { s e. ~P W | E. j j e. s } ) )
15	14	adantl 277	. . . 4 |- ( ( G e. X /\ H e. Y ) -> ( H e. UHGraph <-> F : dom F --> { s e. ~P W | E. j j e. s } ) )
16	15	bicomd 141	. . 3 |- ( ( G e. X /\ H e. Y ) -> ( F : dom F --> { s e. ~P W | E. j j e. s } <-> H e. UHGraph ) )
17	11, 16	sylan9bbr 463	. 2 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( E : dom E --> { s e. ~P V | E. j j e. s } <-> H e. UHGraph ) )
18	5, 17	bitrd 188	1 |- ( ( ( G e. X /\ H e. Y ) /\ ( V = W /\ E = F ) ) -> ( G e. UHGraph <-> H e. UHGraph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   E.wex 1518   e. wcel 2180   {crab 2492   ~Pcpw 3629   dom cdm 4696   -->wf 5290   ` cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  UHGraphcuhgr 15832

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fo 5300  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-uhgrm 15834

This theorem is referenced by: (None)