Theorem uhgreq12g 15930

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 15925	. . . 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 4933	. . . 4 |- ( ( V = W /\ E = F ) -> dom E = dom F )
8		pweq 3655	. . . . . 6 |- ( V = W -> ~P V = ~P W )
9	8	rabeqdv 2796	. . . . 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 5473	. . 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 15925	. . . . 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 2202   {crab 2514   ~Pcpw 3652   dom cdm 4725   -->wf 5322   ` cfv 5326  Vtxcvtx 15866  iEdgciedg 15867  UHGraphcuhgr 15921

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-uhgrm 15923

This theorem is referenced by: (None)