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Theorem inimasn 4741
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 3126 . . 3  |-  ( x  e.  ( ( A
" { C }
)  i^i  ( B " { C } ) )  <->  ( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) ) )
2 elin 3126 . . . . 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 2560 . . . . 5  |-  x  e. 
_V
5 elimasng 4693 . . . . 5  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( ( A  i^i  B
) " { C } )  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
64, 5mpan2 401 . . . 4  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
7 elimasng 4693 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( A " { C } )  <->  <. C ,  x >.  e.  A ) )
84, 7mpan2 401 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( A
" { C }
)  <->  <. C ,  x >.  e.  A ) )
9 elimasng 4693 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( B " { C } )  <->  <. C ,  x >.  e.  B ) )
104, 9mpan2 401 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( B
" { C }
)  <->  <. C ,  x >.  e.  B ) )
118, 10anbi12d 442 . . . 4  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
123, 6, 113bitr4rd 210 . . 3  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  x  e.  ( ( A  i^i  B ) " { C } ) ) )
131, 12syl5rbb 182 . 2  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  x  e.  (
( A " { C } )  i^i  ( B " { C }
) ) ) )
1413eqrdv 2038 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 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2557    i^i cin 2916   {csn 3375   <.cop 3378   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by: (None)
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