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Theorem usgraeq12d 23405
Description: Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
Assertion
Ref Expression
usgraeq12d  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V USGrph  E  <->  W USGrph  F ) )

Proof of Theorem usgraeq12d
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simprr 756 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  E  =  F )
2 dmeq 5124 . . . . 5  |-  ( E  =  F  ->  dom  E  =  dom  F )
32adantl 466 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  dom  E  =  dom  F )
43adantl 466 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  dom  E  =  dom  F
)
5 pweq 3947 . . . . . . 7  |-  ( V  =  W  ->  ~P V  =  ~P W
)
65adantr 465 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ~P V  =  ~P W )
76adantl 466 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  ~P V  =  ~P W )
87difeq1d 3557 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( ~P V  \  { (/) } )  =  ( ~P W  \  { (/) } ) )
9 biidd 237 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( ( # `  x
)  =  2  <->  ( # `
 x )  =  2 ) )
108, 9rabeqbidv 3049 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ( ~P W  \  { (/) } )  |  ( # `  x
)  =  2 } )
111, 4, 10f1eq123d 5720 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  F : dom  F -1-1-> { x  e.  ( ~P W  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
12 isusgra 23393 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
1312adantr 465 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
14 eleq1 2520 . . . . . . . 8  |-  ( V  =  W  ->  ( V  e.  X  <->  W  e.  X ) )
1514biimpd 207 . . . . . . 7  |-  ( V  =  W  ->  ( V  e.  X  ->  W  e.  X ) )
1615adantr 465 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( V  e.  X  ->  W  e.  X ) )
1716com12 31 . . . . 5  |-  ( V  e.  X  ->  (
( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1817adantr 465 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1918imp 429 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  W  e.  X )
20 eleq1 2520 . . . . . . . 8  |-  ( E  =  F  ->  ( E  e.  Y  <->  F  e.  Y ) )
2120biimpd 207 . . . . . . 7  |-  ( E  =  F  ->  ( E  e.  Y  ->  F  e.  Y ) )
2221adantl 466 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( E  e.  Y  ->  F  e.  Y ) )
2322com12 31 . . . . 5  |-  ( E  e.  Y  ->  (
( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2423adantl 466 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2524imp 429 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  F  e.  Y )
26 isusgra 23393 . . 3  |-  ( ( W  e.  X  /\  F  e.  Y )  ->  ( W USGrph  F  <->  F : dom  F -1-1-> { x  e.  ( ~P W  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
2719, 25, 26syl2anc 661 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( W USGrph  F  <->  F : dom  F -1-1-> { x  e.  ( ~P W  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
2811, 13, 273bitr4d 285 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  -> 
( V USGrph  E  <->  W USGrph  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   {crab 2796    \ cdif 3409   (/)c0 3721   ~Pcpw 3944   {csn 3961   class class class wbr 4376   dom cdm 4924   -1-1->wf1 5499   ` cfv 5502   2c2 10458   #chash 12190   USGrph cusg 23385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-usgra 23387
This theorem is referenced by: (None)
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