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Theorem uhgraeq12d 24172
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 5189 . . . . 5  |-  ( E  =  F  ->  dom  E  =  dom  F )
32adantl 466 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  dom  E  =  dom  F )
4 pweq 3996 . . . . . 6  |-  ( V  =  W  ->  ~P V  =  ~P W
)
54difeq1d 3603 . . . . 5  |-  ( V  =  W  ->  ( ~P V  \  { (/) } )  =  ( ~P W  \  { (/) } ) )
65adantr 465 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  ( ~P V  \  { (/) } )  =  ( ~P W  \  { (/) } ) )
71, 3, 6feq123d 5707 . . 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 24163 . . 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 2513 . . . . . . . 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 2513 . . . . . . . 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 24163 . . 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 1381    e. wcel 1802    \ cdif 3455   (/)c0 3767   ~Pcpw 3993   {csn 4010   class class class wbr 4433   dom cdm 4985   -->wf 5570   UHGrph cuhg 24155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-fun 5576  df-fn 5577  df-f 5578  df-uhgra 24157
This theorem is referenced by: (None)
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