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Theorem iscst2 25278
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 25277 . . 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 5924 . . . 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 4922 . 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 6213 . . . 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 4956 . . 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 5991 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 1632    e. wcel 1696   _Vcvv 2801   ~Pcpw 3638   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   csetccst 25275
This theorem is referenced by:  iscst3  25279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-cst 25276
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