Theorem uhgreq12g 15956

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 15951	. . . 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 4935	. . . 4 |- ( ( V = W /\ E = F ) -> dom E = dom F )
8		pweq 3656	. . . . . 6 |- ( V = W -> ~P V = ~P W )
9	8	rabeqdv 2795	. . . . 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 5475	. . 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 15951	. . . . 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 1397   E.wex 1540   e. wcel 2201   {crab 2513   ~Pcpw 3653   dom cdm 4727   -->wf 5324   ` cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  UHGraphcuhgr 15947

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fo 5334  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-uhgrm 15949

This theorem is referenced by: (None)