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Theorem iscst4 25280
Description: The value of the couple  <. A ,  B >. through the derived operation  H (expressed with a union). (Contributed by FL, 31-Dec-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
iscst4  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  = 
U_ x  e.  B  ( A H { x } ) )
Distinct variable groups:    x, A    x, B    x, G    x, M    x, X
Allowed substitution hint:    H( x)

Proof of Theorem iscst4
Dummy variables  a 
y  z 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, 2iscst3 25279 . . 3  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  (
a  e.  ( A H B )  <->  E. y  e.  A  E. x  e.  B  a  =  ( y G x ) ) )
4 simpl1 958 . . . . . . 7  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  G  e.  M )
5 simpl2 959 . . . . . . 7  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  A  e.  ~P X )
6 elpwi 3646 . . . . . . . . . 10  |-  ( B  e.  ~P X  ->  B  C_  X )
763ad2ant3 978 . . . . . . . . 9  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  B  C_  X )
87sselda 3193 . . . . . . . 8  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  x  e.  X )
9 snelpwi 4236 . . . . . . . 8  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
108, 9syl 15 . . . . . . 7  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  { x }  e.  ~P X
)
111, 2iscst3 25279 . . . . . . 7  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  { x }  e.  ~P X )  ->  (
a  e.  ( A H { x }
)  <->  E. y  e.  A  E. z  e.  { x } a  =  ( y G z ) ) )
124, 5, 10, 11syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  (
a  e.  ( A H { x }
)  <->  E. y  e.  A  E. z  e.  { x } a  =  ( y G z ) ) )
13 vex 2804 . . . . . . . 8  |-  x  e. 
_V
14 oveq2 5882 . . . . . . . . 9  |-  ( z  =  x  ->  (
y G z )  =  ( y G x ) )
1514eqeq2d 2307 . . . . . . . 8  |-  ( z  =  x  ->  (
a  =  ( y G z )  <->  a  =  ( y G x ) ) )
1613, 15rexsn 3688 . . . . . . 7  |-  ( E. z  e.  { x } a  =  ( y G z )  <-> 
a  =  ( y G x ) )
1716rexbii 2581 . . . . . 6  |-  ( E. y  e.  A  E. z  e.  { x } a  =  ( y G z )  <->  E. y  e.  A  a  =  ( y G x ) )
1812, 17syl6bb 252 . . . . 5  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  (
a  e.  ( A H { x }
)  <->  E. y  e.  A  a  =  ( y G x ) ) )
1918rexbidva 2573 . . . 4  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( E. x  e.  B  a  e.  ( A H { x } )  <->  E. x  e.  B  E. y  e.  A  a  =  ( y G x ) ) )
20 eliun 3925 . . . 4  |-  ( a  e.  U_ x  e.  B  ( A H { x } )  <->  E. x  e.  B  a  e.  ( A H { x } ) )
21 rexcom 2714 . . . 4  |-  ( E. y  e.  A  E. x  e.  B  a  =  ( y G x )  <->  E. x  e.  B  E. y  e.  A  a  =  ( y G x ) )
2219, 20, 213bitr4g 279 . . 3  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  (
a  e.  U_ x  e.  B  ( A H { x } )  <->  E. y  e.  A  E. x  e.  B  a  =  ( y G x ) ) )
233, 22bitr4d 247 . 2  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  (
a  e.  ( A H B )  <->  a  e.  U_ x  e.  B  ( A H { x } ) ) )
2423eqrdv 2294 1  |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  = 
U_ x  e.  B  ( A H { x } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   {csn 3653   U_ciun 3921   dom cdm 4705   ` cfv 5271  (class class class)co 5874   csetccst 25275
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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