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Theorem inimasn 5429
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 3692 . . 3  |-  ( x  e.  ( ( A
" { C }
)  i^i  ( B " { C } ) )  <->  ( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) ) )
2 elin 3692 . . . . 5  |-  ( <. C ,  x >.  e.  ( A  i^i  B
)  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) )
32a1i 11 . . . 4  |-  ( C  e.  V  ->  ( <. C ,  x >.  e.  ( A  i^i  B
)  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
4 vex 3121 . . . . 5  |-  x  e. 
_V
5 elimasng 5369 . . . . 5  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( ( A  i^i  B
) " { C } )  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
64, 5mpan2 671 . . . 4  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  <. C ,  x >.  e.  ( A  i^i  B ) ) )
7 elimasng 5369 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( A " { C } )  <->  <. C ,  x >.  e.  A ) )
84, 7mpan2 671 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( A
" { C }
)  <->  <. C ,  x >.  e.  A ) )
9 elimasng 5369 . . . . . 6  |-  ( ( C  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( B " { C } )  <->  <. C ,  x >.  e.  B ) )
104, 9mpan2 671 . . . . 5  |-  ( C  e.  V  ->  (
x  e.  ( B
" { C }
)  <->  <. C ,  x >.  e.  B ) )
118, 10anbi12d 710 . . . 4  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  ( <. C ,  x >.  e.  A  /\  <. C ,  x >.  e.  B ) ) )
123, 6, 113bitr4rd 286 . . 3  |-  ( C  e.  V  ->  (
( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) )  <->  x  e.  ( ( A  i^i  B ) " { C } ) ) )
131, 12syl5rbb 258 . 2  |-  ( C  e.  V  ->  (
x  e.  ( ( A  i^i  B )
" { C }
)  <->  x  e.  (
( A " { C } )  i^i  ( B " { C }
) ) ) )
1413eqrdv 2464 1  |-  ( C  e.  V  ->  (
( A  i^i  B
) " { C } )  =  ( ( A " { C } )  i^i  ( B " { C }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   {csn 4033   <.cop 4039   "cima 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018
This theorem is referenced by:  restutopopn  20607  ustuqtop2  20611
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