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Theorem iscst2 25175
Description: The value of the couple  <. A ,  B >. through the derived operation. (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
iscst2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  ran  ( u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
Distinct variable groups:    v, u, A    u, B, v    u, G, v
Allowed substitution hints:    H( v, u)    M( v, u)    X( v, u)

Proof of Theorem iscst2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscst1.1 . . . 4  |-  X  =  dom  dom  G
2 iscst1.2 . . . 4  |-  H  =  ( cset `  G
)
31, 2iscst1 25174 . . 3  |-  ( G  e.  M  ->  H  =  ( x  e. 
~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
433ad2ant1 976 . 2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  H  =  ( x  e. 
~P X ,  y  e.  ~P X  |->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
5 mpt2eq12 5908 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( u  e.  x ,  v  e.  y  |->  ( u G v ) )  =  ( u  e.  A , 
v  e.  B  |->  ( u G v ) ) )
65adantl 452 . . 3  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  (
x  =  A  /\  y  =  B )
)  ->  ( u  e.  x ,  v  e.  y  |->  ( u G v ) )  =  ( u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
76rneqd 4906 . 2  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  (
x  =  A  /\  y  =  B )
)  ->  ran  ( u  e.  x ,  v  e.  y  |->  ( u G v ) )  =  ran  ( u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
8 simp2 956 . 2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  A  e.  ~P X )
9 simp3 957 . 2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  B  e.  ~P X )
10 mpt2exga 6197 . . . 4  |-  ( ( A  e.  ~P X  /\  B  e.  ~P X )  ->  (
u  e.  A , 
v  e.  B  |->  ( u G v ) )  e.  _V )
11103adant1 973 . . 3  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  (
u  e.  A , 
v  e.  B  |->  ( u G v ) )  e.  _V )
12 rnexg 4940 . . 3  |-  ( ( u  e.  A , 
v  e.  B  |->  ( u G v ) )  e.  _V  ->  ran  ( u  e.  A ,  v  e.  B  |->  ( u G v ) )  e.  _V )
1311, 12syl 15 . 2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ran  ( u  e.  A ,  v  e.  B  |->  ( u G v ) )  e.  _V )
144, 7, 8, 9, 13ovmpt2d 5975 1  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  ran  ( u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ 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:  iscst3  25176
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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