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Theorem iinxsng 3856
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 3786 . 2  |-  |^|_ x  e.  { A } B  =  { y  |  A. x  e.  { A } y  e.  B }
2 iinxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2187 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43ralsng 3534 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } y  e.  B  <->  y  e.  C ) )
54abbi1dv 2237 . 2  |-  ( A  e.  V  ->  { y  |  A. x  e. 
{ A } y  e.  B }  =  C )
61, 5syl5eq 2162 1  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   {csn 3497   |^|_ciin 3784
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-sbc 2883  df-sn 3503  df-iin 3786
This theorem is referenced by: (None)
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