Theorem uhgreq12g 16120

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 16115	. . . 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 4960	. . . 4 |- ( ( V = W /\ E = F ) -> dom E = dom F )
8		pweq 3674	. . . . . 6 |- ( V = W -> ~P V = ~P W )
9	8	rabeqdv 2809	. . . . 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 5501	. . 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 16115	. . . . 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 1398   E.wex 1541   e. wcel 2205   {crab 2526   ~Pcpw 3671   dom cdm 4751   -->wf 5350   ` cfv 5354  Vtxcvtx 16056  iEdgciedg 16057  UHGraphcuhgr 16111

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-uhgrm 16113

This theorem is referenced by: (None)