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Theorem iscst1 25174
Description: An operation on the subsets derived from an operation on the elements. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
iscst1.1  |-  X  =  dom  dom  G
iscst1.2  |-  H  =  ( cset `  G
)
Assertion
Ref Expression
iscst1  |-  ( G  e.  A  ->  H  =  ( x  e. 
~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
Distinct variable groups:    u, G, v, x, y    x, X, y
Allowed substitution hints:    A( x, y, v, u)    H( x, y, v, u)    X( v, u)

Proof of Theorem iscst1
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 iscst1.2 . 2  |-  H  =  ( cset `  G
)
2 elex 2796 . . 3  |-  ( G  e.  A  ->  G  e.  _V )
3 dmexg 4939 . . . . 5  |-  ( G  e.  A  ->  dom  G  e.  _V )
4 iscst1.1 . . . . . 6  |-  X  =  dom  dom  G
5 dmexg 4939 . . . . . 6  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
64, 5syl5eqel 2367 . . . . 5  |-  ( dom 
G  e.  _V  ->  X  e.  _V )
7 pwexg 4194 . . . . 5  |-  ( X  e.  _V  ->  ~P X  e.  _V )
83, 6, 73syl 18 . . . 4  |-  ( G  e.  A  ->  ~P X  e.  _V )
9 mpt2exga 6197 . . . 4  |-  ( ( ~P X  e.  _V  /\ 
~P X  e.  _V )  ->  ( x  e. 
~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) )  e. 
_V )
108, 8, 9syl2anc 642 . . 3  |-  ( G  e.  A  ->  (
x  e.  ~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) )  e.  _V )
11 dmeq 4879 . . . . . . . 8  |-  ( g  =  G  ->  dom  g  =  dom  G )
1211dmeqd 4881 . . . . . . 7  |-  ( g  =  G  ->  dom  dom  g  =  dom  dom  G )
1312, 4syl6eqr 2333 . . . . . 6  |-  ( g  =  G  ->  dom  dom  g  =  X )
1413pweqd 3630 . . . . 5  |-  ( g  =  G  ->  ~P dom  dom  g  =  ~P X )
15 simp1 955 . . . . . . . 8  |-  ( ( g  =  G  /\  u  e.  x  /\  v  e.  y )  ->  g  =  G )
1615oveqd 5875 . . . . . . 7  |-  ( ( g  =  G  /\  u  e.  x  /\  v  e.  y )  ->  ( u g v )  =  ( u G v ) )
1716mpt2eq3dva 5912 . . . . . 6  |-  ( g  =  G  ->  (
u  e.  x ,  v  e.  y  |->  ( u g v ) )  =  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) )
1817rneqd 4906 . . . . 5  |-  ( g  =  G  ->  ran  ( u  e.  x ,  v  e.  y  |->  ( u g v ) )  =  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) )
1914, 14, 18mpt2eq123dv 5910 . . . 4  |-  ( g  =  G  ->  (
x  e.  ~P dom  dom  g ,  y  e. 
~P dom  dom  g  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u g v ) ) )  =  ( x  e.  ~P X ,  y  e.  ~P X  |->  ran  (
u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
20 df-cst 25173 . . . 4  |-  cset  =  ( g  e.  _V  |->  ( x  e.  ~P dom  dom  g ,  y  e.  ~P dom  dom  g  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u g v ) ) ) )
2119, 20fvmptg 5600 . . 3  |-  ( ( G  e.  _V  /\  ( x  e.  ~P X ,  y  e.  ~P X  |->  ran  (
u  e.  x ,  v  e.  y  |->  ( u G v ) ) )  e.  _V )  ->  ( cset `  G
)  =  ( x  e.  ~P X , 
y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
222, 10, 21syl2anc 642 . 2  |-  ( G  e.  A  ->  ( cset `  G )  =  ( x  e.  ~P X ,  y  e.  ~P X  |->  ran  (
u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
231, 22syl5eq 2327 1  |-  ( G  e.  A  ->  H  =  ( x  e. 
~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   csetccst 25172
This theorem is referenced by:  iscst2  25175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-cst 25173
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