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Theorem uhgraeq12d 23192
Description: Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
Assertion
Ref Expression
uhgraeq12d  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V UHGrph  E  <->  W UHGrph  F ) )

Proof of Theorem uhgraeq12d
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  E  =  F )
2 dmeq 5035 . . . . 5  |-  ( E  =  F  ->  dom  E  =  dom  F )
32adantl 466 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  dom  E  =  dom  F )
4 pweq 3858 . . . . . 6  |-  ( V  =  W  ->  ~P V  =  ~P W
)
54difeq1d 3468 . . . . 5  |-  ( V  =  W  ->  ( ~P V  \  { (/) } )  =  ( ~P W  \  { (/) } ) )
65adantr 465 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  ( ~P V  \  { (/) } )  =  ( ~P W  \  { (/) } ) )
71, 3, 6feq123d 5544 . . 3  |-  ( ( V  =  W  /\  E  =  F )  ->  ( E : dom  E --> ( ~P V  \  { (/) } )  <->  F : dom  F --> ( ~P W  \  { (/) } ) ) )
87adantl 466 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( E : dom  E --> ( ~P V  \  { (/) } )  <->  F : dom  F --> ( ~P W  \  { (/) } ) ) )
9 isuhgra 23188 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
109adantr 465 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
11 eleq1 2498 . . . . . . . 8  |-  ( V  =  W  ->  ( V  e.  X  <->  W  e.  X ) )
1211biimpd 207 . . . . . . 7  |-  ( V  =  W  ->  ( V  e.  X  ->  W  e.  X ) )
1312adantr 465 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( V  e.  X  ->  W  e.  X ) )
1413com12 31 . . . . 5  |-  ( V  e.  X  ->  (
( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1514adantr 465 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1615imp 429 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  W  e.  X )
17 eleq1 2498 . . . . . . . 8  |-  ( E  =  F  ->  ( E  e.  Y  <->  F  e.  Y ) )
1817biimpd 207 . . . . . . 7  |-  ( E  =  F  ->  ( E  e.  Y  ->  F  e.  Y ) )
1918adantl 466 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( E  e.  Y  ->  F  e.  Y ) )
2019com12 31 . . . . 5  |-  ( E  e.  Y  ->  (
( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2120adantl 466 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2221imp 429 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  F  e.  Y )
23 isuhgra 23188 . . 3  |-  ( ( W  e.  X  /\  F  e.  Y )  ->  ( W UHGrph  F  <->  F : dom  F --> ( ~P W  \  { (/) } ) ) )
2416, 22, 23syl2anc 661 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( W UHGrph  F  <->  F : dom  F --> ( ~P W  \  { (/) } ) ) )
258, 10, 243bitr4d 285 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V UHGrph  E  <->  W UHGrph  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3320   (/)c0 3632   ~Pcpw 3855   {csn 3872   class class class wbr 4287   dom cdm 4835   -->wf 5409   UHGrph cuhg 23184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-fun 5415  df-fn 5416  df-f 5417  df-uhgra 23185
This theorem is referenced by: (None)
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