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Theorem inimasn 4771
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 3156 . . 3  |-  ( x  e.  ( ( A
" { C }
)  i^i  ( B " { C } ) )  <->  ( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) ) )
2 elin 3156 . . . . 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 2605 . . . . 5  |-  x  e. 
_V
5 elimasng 4723 . . . . 5  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( ( A  i^i  B
) " { C } )  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
64, 5mpan2 416 . . . 4  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
7 elimasng 4723 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( A " { C } )  <->  <. C ,  x >.  e.  A ) )
84, 7mpan2 416 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( A
" { C }
)  <->  <. C ,  x >.  e.  A ) )
9 elimasng 4723 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( B " { C } )  <->  <. C ,  x >.  e.  B ) )
104, 9mpan2 416 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( B
" { C }
)  <->  <. C ,  x >.  e.  B ) )
118, 10anbi12d 457 . . . 4  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
123, 6, 113bitr4rd 219 . . 3  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  x  e.  ( ( A  i^i  B ) " { C } ) ) )
131, 12syl5rbb 191 . 2  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  x  e.  (
( A " { C } )  i^i  ( B " { C }
) ) ) )
1413eqrdv 2080 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2602    i^i cin 2973   {csn 3406   <.cop 3409   "cima 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by: (None)
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