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Theorem iinxsng 4353
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 4276 . 2  |-  |^|_ x  e.  { A } B  =  { y  |  A. x  e.  { A } y  e.  B }
2 iinxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2474 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43ralsng 4009 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } y  e.  B  <->  y  e.  C ) )
54abbi1dv 2542 . 2  |-  ( A  e.  V  ->  { y  |  A. x  e. 
{ A } y  e.  B }  =  C )
61, 5syl5eq 2457 1  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   {cab 2389   A.wral 2756   {csn 3974   |^|_ciin 4274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-ral 2761  df-v 3063  df-sbc 3280  df-sn 3975  df-iin 4276
This theorem is referenced by:  polatN  32961
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