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Theorem iinxsng 3978
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 3908 . 2  |-  |^|_ x  e.  { A } B  =  { y  |  A. x  e.  { A } y  e.  B }
2 iinxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2350 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43ralsng 3672 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } y  e.  B  <->  y  e.  C ) )
54abbi1dv 2399 . 2  |-  ( A  e.  V  ->  { y  |  A. x  e. 
{ A } y  e.  B }  =  C )
61, 5syl5eq 2327 1  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   {csn 3640   |^|_ciin 3906
This theorem is referenced by:  splintx  25049  polatN  30120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992  df-sn 3646  df-iin 3908
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