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Theorem inimasn 5266
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 3551 . . 3  |-  ( x  e.  ( ( A
" { C }
)  i^i  ( B " { C } ) )  <->  ( x  e.  ( A " { C } )  /\  x  e.  ( B " { C } ) ) )
2 elin 3551 . . . . 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 2987 . . . . 5  |-  x  e. 
_V
5 elimasng 5207 . . . . 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 5207 . . . . . 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 5207 . . . . . 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 2441 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 1369    e. wcel 1756   _Vcvv 2984    i^i cin 3339   {csn 3889   <.cop 3895   "cima 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865
This theorem is referenced by:  restutopopn  19825  ustuqtop2  19829
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