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Theorem iinxsng 3944
Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Hypothesis
Ref Expression
iinxsng.1  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iinxsng  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem iinxsng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-iin 3874 . 2  |-  |^|_ x  e.  { A } B  =  { y  |  A. x  e.  { A } y  e.  B }
2 iinxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2240 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43ralsng 3621 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } y  e.  B  <->  y  e.  C ) )
54abbi1dv 2290 . 2  |-  ( A  e.  V  ->  { y  |  A. x  e. 
{ A } y  e.  B }  =  C )
61, 5eqtrid 2215 1  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   {csn 3581   |^|_ciin 3872
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956  df-sn 3587  df-iin 3874
This theorem is referenced by: (None)
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