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Theorem iscst4 25177
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 25176 . . 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 3633 . . . . . . . . . 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 3180 . . . . . . . 8  |-  ( ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  /\  x  e.  B )  ->  x  e.  X )
9 snelpwi 4220 . . . . . . . 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 25176 . . . . . . 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 2791 . . . . . . . 8  |-  x  e. 
_V
14 oveq2 5866 . . . . . . . . 9  |-  ( z  =  x  ->  (
y G z )  =  ( y G x ) )
1514eqeq2d 2294 . . . . . . . 8  |-  ( z  =  x  ->  (
a  =  ( y G z )  <->  a  =  ( y G x ) ) )
1613, 15rexsn 3675 . . . . . . 7  |-  ( E. z  e.  { x } a  =  ( y G z )  <-> 
a  =  ( y G x ) )
1716rexbii 2568 . . . . . 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 2560 . . . 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 3909 . . . 4  |-  ( a  e.  U_ x  e.  B  ( A H { x } )  <->  E. x  e.  B  a  e.  ( A H { x } ) )
21 rexcom 2701 . . . 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 2281 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 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   {csn 3640   U_ciun 3905   dom cdm 4689   ` cfv 5255  (class class class)co 5858   csetccst 25172
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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