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Theorem usgraeq12d 24025
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 5194 . . . . 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 4006 . . . . . . 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 3614 . . . 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 3101 . . 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 5802 . 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 24007 . . 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 2532 . . . . . . . 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 2532 . . . . . . . 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 24007 . . 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 1374    e. wcel 1762   {crab 2811    \ cdif 3466   (/)c0 3778   ~Pcpw 4003   {csn 4020   class class class wbr 4440   dom cdm 4992   -1-1->wf1 5576   ` cfv 5579   2c2 10574   #chash 12360   USGrph cusg 23993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-usgra 23996
This theorem is referenced by: (None)
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