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Theorem usgraeq12d 24779
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 758 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  E  =  F )
2 dmeq 5024 . . . . 5  |-  ( E  =  F  ->  dom  E  =  dom  F )
32adantl 464 . . . 4  |-  ( ( V  =  W  /\  E  =  F )  ->  dom  E  =  dom  F )
43adantl 464 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  dom  E  =  dom  F
)
5 pweq 3958 . . . . . . 7  |-  ( V  =  W  ->  ~P V  =  ~P W
)
65adantr 463 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ~P V  =  ~P W )
76adantl 464 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  ~P V  =  ~P W )
87difeq1d 3560 . . . 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 3054 . . 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 5794 . 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 24761 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
1312adantr 463 . 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 2474 . . . . . . . 8  |-  ( V  =  W  ->  ( V  e.  X  <->  W  e.  X ) )
1514biimpd 207 . . . . . . 7  |-  ( V  =  W  ->  ( V  e.  X  ->  W  e.  X ) )
1615adantr 463 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( V  e.  X  ->  W  e.  X ) )
1716com12 29 . . . . 5  |-  ( V  e.  X  ->  (
( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1817adantr 463 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  W  e.  X ) )
1918imp 427 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  W  e.  X )
20 eleq1 2474 . . . . . . . 8  |-  ( E  =  F  ->  ( E  e.  Y  <->  F  e.  Y ) )
2120biimpd 207 . . . . . . 7  |-  ( E  =  F  ->  ( E  e.  Y  ->  F  e.  Y ) )
2221adantl 464 . . . . . 6  |-  ( ( V  =  W  /\  E  =  F )  ->  ( E  e.  Y  ->  F  e.  Y ) )
2322com12 29 . . . . 5  |-  ( E  e.  Y  ->  (
( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2423adantl 464 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  =  W  /\  E  =  F )  ->  F  e.  Y ) )
2524imp 427 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( V  =  W  /\  E  =  F ) )  ->  F  e.  Y )
26 isusgra 24761 . . 3  |-  ( ( W  e.  X  /\  F  e.  Y )  ->  ( W USGrph  F  <->  F : dom  F -1-1-> { x  e.  ( ~P W  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
2719, 25, 26syl2anc 659 . 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 367    = wceq 1405    e. wcel 1842   {crab 2758    \ cdif 3411   (/)c0 3738   ~Pcpw 3955   {csn 3972   class class class wbr 4395   dom cdm 4823   -1-1->wf1 5566   ` cfv 5569   2c2 10626   #chash 12452   USGrph cusg 24747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-usgra 24750
This theorem is referenced by: (None)
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