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Theorem inimasn 4924
Description: The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimasn  |-  ( C  e.  V  ->  (
( A  i^i  B
) " { C } )  =  ( ( A " { C } )  i^i  ( B " { C }
) ) )

Proof of Theorem inimasn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3227 . . 3  |-  ( x  e.  ( ( A
" { C }
)  i^i  ( B " { C } ) )  <->  ( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) ) )
2 elin 3227 . . . . 5  |-  ( <. C ,  x >.  e.  ( A  i^i  B
)  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) )
32a1i 9 . . . 4  |-  ( C  e.  V  ->  ( <. C ,  x >.  e.  ( A  i^i  B
)  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
4 vex 2661 . . . . 5  |-  x  e. 
_V
5 elimasng 4875 . . . . 5  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( ( A  i^i  B
) " { C } )  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
64, 5mpan2 419 . . . 4  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
7 elimasng 4875 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( A " { C } )  <->  <. C ,  x >.  e.  A ) )
84, 7mpan2 419 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( A
" { C }
)  <->  <. C ,  x >.  e.  A ) )
9 elimasng 4875 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( B " { C } )  <->  <. C ,  x >.  e.  B ) )
104, 9mpan2 419 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( B
" { C }
)  <->  <. C ,  x >.  e.  B ) )
118, 10anbi12d 462 . . . 4  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
123, 6, 113bitr4rd 220 . . 3  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  x  e.  ( ( A  i^i  B ) " { C } ) ) )
131, 12syl5rbb 192 . 2  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  x  e.  (
( A " { C } )  i^i  ( B " { C }
) ) ) )
1413eqrdv 2113 1  |-  ( C  e.  V  ->  (
( A  i^i  B
) " { C } )  =  ( ( A " { C } )  i^i  ( B " { C }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658    i^i cin 3038   {csn 3495   <.cop 3498   "cima 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by: (None)
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