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Theorem iinxsng 4349
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 4272 . 2  |-  |^|_ x  e.  { A } B  =  { y  |  A. x  e.  { A } y  e.  B }
2 iinxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2534 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43ralsng 3997 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } y  e.  B  <->  y  e.  C ) )
54abbi1dv 2591 . 2  |-  ( A  e.  V  ->  { y  |  A. x  e. 
{ A } y  e.  B }  =  C )
61, 5syl5eq 2517 1  |-  ( A  e.  V  ->  |^|_ x  e.  { A } B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   {csn 3959   |^|_ciin 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-ral 2761  df-v 3033  df-sbc 3256  df-sn 3960  df-iin 4272
This theorem is referenced by:  polatN  33567
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